In this work a novel method to solve the direct dynamic problem of robot manipulators based on the dynamic Gibbs Appell equations, is described. The method is based on the obtention of the system inertia matrix by means of the Gibbs function hessian. To obtain this matrix, an efficient O(n2) recursive algorithm in tensor notation, which is used in the elaboration of a FORTRAN program, is proposed. With this program the movement of a robot PUMA is simulated. The simulation results are compared with those obtained using other well-known methods, concluding that the proposed method presents a good numeric behavior.