The complexities in the dynamic model of the legged robots make it necessary to utilize model-free controllers in the task of trajectory tracking. In This paper, an adaptive transpose Jacobian approach is proposed to deal with the dynamic model complexity, which utilizes an adaptive PI-algorithm to adjust the control gains. The performance of the proposed control algorithm is compared with the conventional transpose Jacobian and sliding mode control algorithms and evaluated by the root mean square of the errors and control input energy criteria. In order to appraise the effectiveness of the proposed control system, simulations are carried out in MATLAB/Simulink software for a quadruped robot leg for semi-elliptical path tracking. The obtained results show that the proposed adaptive transpose Jacobian reduces the overshoot and root mean square of the errors and at the same time, decreases the control input energy. Moreover, transpose Jacobin and adaptive transpose Jacobian are more robust to changes in initial conditions compared to the conventional sliding mode control. Furthermore, sliding mode control performs well up to 20% uncertainties in the parameters due to its model-based nature, whereas the transpose Jacobin and the proposed adaptive transpose Jacobian algorithms show promising results even in higher mass uncertainties.