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This work presents a solution to the adaptive tracking control of Euler Lagrange systems with guaranteed tracking and parameter estimation error convergence. Specifically a concurrent learning based update rule fused by the filtered version of the desired system dynamics in conjunction with a desired state based regression matrix has been utilized to ensure that both the position tracking error and parameter estimation error terms converge to origin exponentially. As the regression matrix used in proposed controller makes use of the desired versions of the system states, an initial, sufficiently exciting memory stack can be formed from the knowledge of the desired system trajectory a priori, thus removing the initial excitation condition required for the previously proposed concurrent learning based controllers in the literature. The output feedback versions of the proposed method where only the position measurements are available for the controller design, (for both gradient and composite type adaptions) are also presented in order to illustrate the modularity of the proposed method. The stability and boundedness of the closed loop signals for all the proposed controllers are ensured via Lyapunov based analysis. %Trajectory tracking control of a class of fully actuated Euler Lagrange systems is considered in this work. System dynamics is considered to be subject to parametric uncertainties and on--line identification uncertain model parameters is also aimed. When compared with the relevant past research, via a novel approach, desired states are proposed to be used in forming the regression matrix and a desired compensation based concurrent learning type adaptive update rule is designed. Via utilizing novel Lyapunov analysis, semi--global exponential convergence of both tracking and parameter identification error to the origin is ensured.