A method for the estimation of kinematics of a system of rigid bodies connected by three degrees of freedom rotational joints using position measurements is introduced. In the proposed approach, system kinematics are computed from experimental measurements while preserving important physical and kinematic properties. These properties include system integrity, i.e., preserving interconnections between the bodies, and the entire system dynamic properties, namely, center of mass kinematics and its angular momentum. The computational procedure consists in solving a sequence of optimizations of appropriately formulated objective functions that incorporate the preservation of physical and kinematic properties by employing the penalty function approach. The configuration of the segment kinematics of the system is computed via a quaternion parametrization of orientation that leads to an efficient computation procedure. The sequence of optimization problems is solved using an embedded iteration process. Two studies are presented to demonstrate the performance of the proposed approach: estimations of the kinematics of a simulated three-link model and of an experimentally measured 3D motion of human body during flight phase of a jump. The results of the two studies indicate fast convergence of the algorithm to an optimal solution while accurately satisfying the imposed the constraints.