1. A method of operating a surgical system, the method comprising: measuring a position of a surgical instrument, the surgical instrument being included in a mechanical assembly having a plurality of joints and first degrees of freedom, the position of the surgical instrument being measured for each of second degrees of freedom of the surgical instrument to determine position measurements corresponding to the second degrees of freedom; estimating, with at least one computer, a position of each joint of the plurality of joints, wherein estimating the position of each joint of the plurality of joints includes applying the position measurements to a kinematic model of the mechanical assembly, the kinematic model having third degrees of freedom, the third degrees of freedom including a corresponding degree of freedom for each of the first degrees of freedom and further including at least one specific degree of freedom that is missing from the first degrees of freedom, the application of the position measurements to the kinematic model including a forward-kinematic simulation of each of the third degrees of freedom for at least one time step, and the estimating of the position of each joint of the plurality of joints including a discarding of values for the at least one specific degree of freedom from a result of the forward-kinematic simulation; and controlling the mechanical assembly based on the estimated positions of the plurality of joints.