This paper presents a novel randomized motion planner for robots that must achieve a specified goal under kinematic and/or dynamic motion constraints while avoiding collision with moving obstacles. The planner encodes the motion constraints using a control system and samples the robot's state×time space by randomly selecting control inputs and integrating the equations of motion. This results in a probabilistic roadmap of sampled state×time points, called milestones, connected by short admissible trajectories. The planner does not precompute the roadmap but generates a new roadmap for each planning query, connecting an initial and goal state×time point. The paper provides a detailed analysis of the planner's convergence rate, showing that if the state×time space satisfies a geometric property called expansiveness, the planner is guaranteed to find a trajectory when one exists, with probability quickly converging to 1 as the number of milestones increases. The planner was tested in simulated environments and on a real robot, where a vision module estimated obstacle motions before planning. The planner was also extended to handle time delays and uncertainties inherent in integrated robotic systems interacting with the physical world.This paper presents a novel randomized motion planner for robots that must achieve a specified goal under kinematic and/or dynamic motion constraints while avoiding collision with moving obstacles. The planner encodes the motion constraints using a control system and samples the robot's state×time space by randomly selecting control inputs and integrating the equations of motion. This results in a probabilistic roadmap of sampled state×time points, called milestones, connected by short admissible trajectories. The planner does not precompute the roadmap but generates a new roadmap for each planning query, connecting an initial and goal state×time point. The paper provides a detailed analysis of the planner's convergence rate, showing that if the state×time space satisfies a geometric property called expansiveness, the planner is guaranteed to find a trajectory when one exists, with probability quickly converging to 1 as the number of milestones increases. The planner was tested in simulated environments and on a real robot, where a vision module estimated obstacle motions before planning. The planner was also extended to handle time delays and uncertainties inherent in integrated robotic systems interacting with the physical world.