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This paper presents a strategy for planning robot motions in dynamic, uncertain environments (DUEs). Successful and efficient robot operation in such environments requires reasoning about the future evolution and uncertainties of the states of the moving agents and obstacles. A novel procedure to account for future information gathering (and the quality of that information) in the planning process is presented. To approximately solve the stochastic dynamic programming problem that is associated with DUE planning, we present a partially closed-loop receding horizon control algorithm whose solution integrates prediction, estimation, and planning while also accounting for chance constraints that arise from the uncertain locations of the robot and obstacles. Simulation results in simple static and dynamic scenarios illustrate the benefit of the algorithm over classical approaches. The approach is also applied to more complicated scenarios, including agents with complex, multimodal behaviors, basic robot-agent interaction, and agent information gathering.