Methods, apparatus, and systems for operating a surgical system. In accordance with a method, a position of a surgical instrument is measured, the surgical instrument being included in a mechanical assembly having a plurality of joints and a first number of degrees of freedom, the position of the surgical instrument being measured for each of a second number of degrees of freedom of the surgical instrument. The method further includes estimating a position of each of the joints, where estimating the position of each joint includes applying the position measurements to at least one kinematic model of the mechanical assembly, the kinematic model having a third number of degrees of freedom greater than the first number of degrees of freedom. The method further includes controlling the mechanical assembly based on the estimated position of the joints.