Formulating the suitable mathematical models and deriving the efficient algorithms are very crucial for simplifying the complexity of the kinematics problems in robotics. Forward and inverse kinematics of industrial robot manipulators is generally performed using Denevit and Hartenberg method. In this paper, the algebra based on the exponential rotational matrices is used as an alternative method for driving the direct and inverse kinematics of the 6-DOF industrial robot manipulators with Euler wrist. The main advantage of this method arises from the explicit physical interpretation of the mechanism without deriving DH table. The exponential rotational matrices can also be easily implemented in the programming of robot manipulators. The RS and NS robot manipulators are given as examples to demonstrate the effectiveness of the employed method.