Toward force control of a quadrotor UAV in SE(3)

Two novel model-free second order sliding mode controllers are proposed for the constrained underactuated position and orientation dynamic model of the quadrotor, i.e., considering contact wrench, based on spring-like contact force model. The main theorem establishes conditions for the closed-loop system to guarantee semi-global exponential and robust tracking of position and admissible contact forces, with zero yaw, by exploiting the solution in SE(3). It is proved an integral sliding mode for all time and for any initial condition at a quaternion-based sliding surface. This yields a causal and analytical computation of desired angular velocity in terms of position control, without involving derivatives of the state. A second theorem is derived for terminal stability with desired finite time convergence. This in turns produces three results, a) a singularity-free representation under proper and easy to meet initialization; b) stable transition from free flying to constrained motion, and c) realization of the virtual position controller due to finite time convergence of angular tracking errors. It is noted that there arises an evident physical limit for force interaction along underactuated directions: the more interaction force along the underactuated axes, the more roll and pitch angles are needed the less thrust that can be produced to sustain the UAV in the air. Comprehensive simulation study is discussed under various flying conditions, and finally, the explicit (active) force control, based on a differential algebraic model, is briefly addressed.