We study the problem of generating continuous steering control for robots to completely cover a bounded region over a finite time. First, we pack the area by disks of minimum number, and then a neural network based path planning method is adopted to generate complete coverage paths avoiding collisions with stationary and moving objects. Smooth trajectory is developed using parametric polynomial approximation, and continuous steering control is designed to track the trajectory exactly. The algorithm works for car-like robots which have nonholonomic motion constraints, and the control inputs are represented by analytic functions of trajectory parameters. The design is finally extended to cooperative sweeping of multiple robots. Satisfactory performances are observed in simulations