Input¬output approximation of spacecraft motion is convenient and necessary in many situations. For a rigid¬body spacecraft, this process is simple because the system is governed by a set of equations that is linear in the system parameters. However, the combination of a flexible appendage and a rigid hub adds complexity by increasing the degrees of freedom and by introducing nonlinear coupling between the hub and appendage. Assumed Modes is one technique for modeling flexible body motion. Traditional Assumed Modes uses a set of linear assumed modes, but when dealing with rotating flexible systems, a modification of this method allows for the use of quadratic assumed modes. The quadratic assumed model provides an increased level of sophistication, but the derivation still neglects a set of higher¬order terms. This work develops the nonlinear equations of motion that arise from retaining these nonlinear, higher¬order terms. Simulation results reveal that the inclusion of these terms noticeably changes the motion of the system. Once the equations of motion have been developed, focus turns to the input¬output mapping of a system that is simulated using this set of equations. Approxi¬mating linear systems is straightforward, and many methods exist that can success¬fully perform this function. On the other hand, few approximation methods exist for nonlinear systems. Researchers at Texas A&M University recently developed one such method that obtains a linear estimation and then uses an adaptive polynomial estimation method to compensate for the disparity between that estimate and the true measurements. This research includes an in¬depth investigation of this nonlinear approximation technique. Finally, these two major research thrusts are combined, and input¬output approx¬imation is performed on the nonlinear rotational spacecraft model developed herein. The results of this simulation show that the nonlinear method holds a significant advantage over a traditional linear method in certain situations. Specifically, the nonlinear algorithm provides superior approximation for systems with nonzero natu¬ral frequencies. For the algorithm to be successful when rigid¬body modes are present, the system motion must be persistently exciting. This research is an important first step toward developing a nonlinear parameter identification algorithm.