Attitude reconstruction of a free falling sphere for the experiment Multiple Spheres for Characterization of Atmosphere Temperatures (MUSCAT) is studied in this paper. The attitude dynamics is modeled through Euler's rotational equations of motion. To estimate uncertain parameters in this model such as the matrix of inertia and the lever arm for the dynamic pressure with respect to the center of mass, the dynamics reconstruction can be formulated as an optimization problem. The goal is to minimize the deviation between the measurements and the propagation from the system equations. This approach was tested against a couple of flight data sets which correspond to different periods of time. The result is very reasonable compared to the laboratory test. The estimate can be improved further through allowing drag coefficients variable and taking advantage of measurements from a magnetometer in numerical calculation.