Robots with flexible joints are gaining importance in areas such as collaborative robots (cobots), exoskeletons, and prostheses. They are meant to directly interact with humans, and the emphasis in their construction is not on precision but rather on weight reduction and soft interaction with humans. Well-known rigid robot control strategies are not valid in this area, so new control methods have been proposed to deal with the complexity introduced by elasticity. Some of these methods are seldom used and are unknown to most of the academic community. After selecting the methods, we carried out a comprehensive comparative study of algorithms: simple gravity compensation (Sgc), the singular perturbation method (Spm), the passivity-based approach (Pba), backstepping control design (Bcd), and exact gravity cancellation (Egc). We modeled these algorithms using MATLAB and simulated them for different stiffness levels. Furthermore, their practical implementation was analyzed from the perspective of the magnitudes to be measured and the computational costs of their implementation. In conclusion, the Sgc method is a fast and affordable solution if joint stiffness is relatively high. If good performance is necessary, the Pba is the best option.