Navigation in dynamic or unknown environment is a challenge for autonomous vehicles because of the limited range of sensors and not accurate maps. In this paper, we analyze three conditions on the unobserved and uncertainty environment during navigation. They are "known space", "free space", and "unknown space". For the dynamic environment, we derive an algorithm to correct the false obstacles in the map when a conventional path planning is stuck. For the unknown environment, we derive novel algorithms and compare them with the classical approaches under free space condition. Finally we use Monte Carlo method to evaluate the performance of these algorithms. Experimental results show that our conditions based algorithms are better than the others.