<p>Inspired by animals’ righting reflex behavior, a free-fall robotic system with zero angular momentum can be controlled to orient one or more of its bodies with respect to the environment by periodically moving some of its joints. However, considering the complex nonlinear dynamics and nonholonomic constraints, planning such periodical closed-loop joint motion trajectories for achieving a required orientation of one specific body (such as the head or base of the robot) remains a challenging robot dynamics and control problem. In this paper, we propose a deep learning-based method to address the challenge. First, the method employs a reconstruction loss for training with non-gradient simulation data, which increases the solution’s accuracy. Second, it approximates general inverse solutions of the system and expands the reachable rotation in a single motion to reduce the repetition of periodical motions. Third, with the generalization of neural networks, the method can be applied to systems with unknown physical properties and can plan the needed periodical motions in real time after training. As an example, the method has been implemented on a squirrel-like quadruped multibody system, and its simulation results are compared with other methods as the baseline solution.</p>