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In mobile robot applications such as cleaning and security patrolling, a fundamentally important problem is to design feasible trajectories and steering control so that the robot moves collision-free and covers all the points (in its sensor/effector range) in a dynamic and uncertain environment. We formulate such a problem and propose constructive algorithms in sequential modules to solve it. First, a minimum-area rectangle is placed encasing the boundary of the set to be covered. Second, minimum number of circles of the radius of coverage range are placed to completely cover the rectangle. Third, a patrolling path is searched along the boundary of the set in a spital. Feasible trajectories are then designed to account for the nonholonomic kinematics of the robot and to avoid collisions from the dynamic obstacles detected by the robot onboard sensors. Since analytic solutions are given in generating feasible trajectories, the algorithm can be implemented in real time.