A seemingly simple rigid-gyro-motion problem involves an overlooked aspect in university physics teaching, namely, the calculation of angular momentum when rigid bodies rotate simultaneously around mutually perpendicular axes. Starting from the solution of question, this paper elaborates the complex relationship of angular momentum of rigid-body (gyro) rotation, and derives the correct angular momentum transformation relationship of simple and constrained gyroscope. At the same time, it shows that, this kind of problems can be greatly simplified when selecting the appropriate reference frame.