Integrated Guidance and Feedback Control of Underactuated Robotics System in SE(3)

Abstract

An integrated guidance and feedback control scheme for steering an underactuated vehicle through desired waypoints in three-dimensional space, is developed here. The underactuated vehicle is 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 multirotor unmanned aerial vehicles (UAVs), maneuverable robotic vehicles, etc. The guidance problem is developed on the special Euclidean group of rigid body motions, SE(3), in the framework ofgeometric mechanics, which represents the vehicle dynamics globally on this configuration manifold. The integrated guidance and control algorithm selects the desired trajectory for the translational motion that passes through the given waypoints, and the desired trajectory for the attitude based on the desired 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 account 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 of the feedback system over its state space is shown analytically and verified through numerical simulations.