The aim of this study is to eliminate the force indeterminacy in a Hooke joint mechanism. It is known that the Hooke joint mechanism is an overconstrained mechanism, so it is impossible to determine some of the unknown reaction forces and moments with the available number of force and moment equations. With the aid of extra joint additions and joint modifications as proposed in this study, it becomes possible to eliminate this force indeterminacy. The addition of extra joints increases the number of links; therefore the physical structure of the Hooke joint mechanism is altered. However, it is required to make a general and detailed singularity analysis to investigate the validity of all possible arrangements of joint additions and modifications. For this purpose, a general position and velocity analysis has been made to search for the singularities of the modified Hooke joint mechanisms.