Aerial Robots Carrying Flexible Cables: Dynamic Shape Optimal Control via Spectral Method Model

Abstract

In this work, we present a model-based optimal boundary control design for an aerial robotic system composed of a quadrotor carrying a flexible cable. The whole system is modeled by partial differential equations (PDEs) combined with boundary conditions described by ordinary differential equations (ODEs). The proper orthogonal decomposition (POD) method is adopted to project the original infinite-dimensional system on a finite low-dimensional space spanned by orthogonal basis functions. Based on such a reduced order model, nonlinear model predictive control (NMPC) is implemented online to realize both position and shape trajectory tracking of the flexible cable in an optimal predictive fashion. The proposed POD-based reduced modeling and optimal control paradigms are verified in simulation using an accurate high-dimensional FDM-based model and experimentally using a real quadrotor and a cable. The results show the viability of the POD-based predictive control approach (allowing closing the control loop on the full system state) and its superior performance compared to an optimally tuned PID controller (allowing closing the control loop on the quadrotor state only).

Figures (24)

• Figure 1: (a): 10 cm quadrotor equipped attached to a cable, which is measured through reflective tape; (b): schematics of the quadrotor-cable system and its configuaration.
• Figure 2: Control architecture of Quadrotor-cable system (in red, in the center) connected to the high fidelity PDE-ODE simulator (in white, on the right ) and to the cable trajectory planner (in blue, on the left).
• Figure 3: Cable POD modes (a) Displacement for first five POD modes of the cable (b) Energy distribution for each POD mode.
• Figure 4: Snapshots of the cable simulated using FDM (red) and ROM (blue) with different order $K$.
• Figure 5: Comparation between FDM model and ROMs with different truncated order $K$.