Time-optimal motion planning of cable-suspended payload quadcopters using spatial reformulation

Cable-suspended payload quadcopters (CSPQ) have gained significant attention due to their mechanical simplicity and agility for various applications. However, the presence of a slung payload introduces complex dynamics, making the system highly underactuated and challenging to control. An accurate and intuitive representation of the safety region is also critical for collision avoidance as the operating environment is always complex and cluttered. This paper proposes a unified hierarchical framework for time-optimal motion planning and trajectory tracking of CSPQ systems, built upon a novel spatial reformulation method leveraging the Gravity-Normal (GN) frame. As an alternative to conventional reformulation approaches, this method transforms the system positions into path-parametric coordinates with clear physical interpretation and no singularities, thereby simplifying the representation of spatial constraints. The hierarchical framework comprises an offline module for global time-optimal trajectory generation using the direct multiple shooting method and an online module for real-time trajectory tracking through nonlinear model predictive control (NMPC). Simulation experiments demonstrated the effectiveness of the proposed framework in achieving time-optimal motion while suppressing payload oscillations within acceptable limits.