Integrated guidance and nonlinear feedback control of underactuated unmanned aerial vehicles in SE(3)

An integrated guidance and feedback control scheme for steering underactuated vehicles through desired waypoints in three-dimensional space, is developed here. The underactuated vehicles have fewer control inputs than the number of configuration variables and modeled as a rigid body with four control inputs. These control inputs actuate the three degrees of freedom of rotational motion and one degree of freedom of translational motion in a vehicle body-fixed coordinate frame. This actuation model is appropriate for a wide range of underactuated vehicles including spacecraft with internal attitude actuators, vertical take-off and landing (VTOL) aircraft, fixed-wing and multirotor unmanned aerial vehicles (UAVs), etc. Here, the guidance problem is developed on the special Euclidean group of rigid body motions, SE(3), in the framework of geometric mechanics, which represents the vehicle dynamics globally on this configuration manifold. The integrated guidance and control algorithm selects a desired trajectory for the translational motion that passes through the given waypoints, and a desired trajectory for the attitude based on the de- sired thrust direction to achieve the translational motion trajectory. A feedback control law is then obtained to steer the underactuated vehicle towards the desired trajectories in translation and rotation. This integrated guidance and control scheme takes into ac- count known bounds on control inputs and generates a trajectory that is continuous and at least twice differentiable, which can be implemented with continuous and bounded control inputs. The integrated guidance and feedback control scheme is applied to an underactuated quadcopter UAV to autonomously generate a trajectory through a series of given waypoints in SE(3) and track the desired trajectory in finite time. The overall stability analysis of the feedback system is addressed. Discrete time models for the dynamics and control schemes of the UAV are obtained in the form of Lie group variational integrators using the discrete Lagrange-d'Alembert principle. Almost global asymptotic stability and domain of convergence of the overall feedback system over the state space is demonstrated both analytically and through numerical simulations.