Solving the robot arm path-planning problem can be considered as a significant aspect in the robot navigation that should guarantees a free collide with obstacles in order to construct the best path. In this paper, the shortest path and trajectory planning of two-link robot arm with 2-DOF in the 2-D static known environment has been analyzed. This analysis deals with three main problems. The first one concerns with the construction of the free Cartesian space by analyzing the inverse kinematic solutions, which guarantees a free collision path planning. The second problem focuses on generating the shortest path that satisfied the aims of motion by applying the D* algorithm. The third problem concerns with selecting the specified number of intermediate via-points and attaining the corresponding smooth trajectory through using Fifth-order polynomial equations. The obtained results illustrate that the free Cartesian space ensures a free collision path and trajectory planning.