Stability of Helicopters in Compliant Contact Under PD-PID Control

Aerial vehicles are difficult to stabilize, especially when acted upon by external forces. A hovering vehicle interacting with objects and surfaces must be robust to contact forces and torques transmitted to the airframe. These produce coupled dynamics that are distinctly different from those of free flight. While external contact is generally avoided, extending aerial robot functionality to include contact with the environment during flight opens up new and useful areas such as perching, object grasping, and manipulation. These mechanics may be modeled as elastic couplings between the aircraft and the ground, represented by springs in R-3 x SO(3). We show that proportional derivative and proportional integral derivative (PID) attitude and position controllers that stabilize a rotorcraft in free flight will also stabilize the aircraft during contact for a range of contact displacements and stiffnesses. Simulation of the coupled aircraft dynamics demonstrates stable and unstable modes of the system. We find analytical measures that predict the stability of these systems and consider, in particular, the planar system in which the contact point is directly beneath the rotor. We show through explicit solution of the linearized system that the planar dynamics of the object-helicopter system in vertical, horizontal, and pitch motion around equilibrium remain stable, within a range of contact stiffnesses, under unmodified PID attitude control. Flight experiments with a small-scale PID-stabilized helicopter fitted with a compliant gripper for capturing objects affirm our model's stability predictions.