This paper presents an efficient algorithm for online avoidance of static obstacles that accounts for robot dynamics and actuator constraints. The robot trajectory (path and speed) is generated incrementally by avoiding obstacles optimally one at a time, thus reducing the original problem from one that avoids m obstacles to m simpler problems that avoid one obstacle each. The computational complexity of this planner is therefore linear in the number of obstacles, instead of the typical exponential complexity of traditional geometric planners. This approach is quite general and applicable to any cost function and to any robot dynamics; it is treated here for minimum time motions, a planar point mass robot, and circular obstacles. Numerical experiments demonstrate the algorithm for very cluttered environments (70 obstacles) and narrow passages. Upper and lower bounds on the motion time and on the path length were derived as functions of the Euclidean distance between the end points and the average obstacle size. Comparing a kinematic version of this algorithm to the RRT and RRT* planners showed significant advantages over both planners. The algorithm was demonstrated on an experimental mobile robot moving at high speeds in a known environment.