A Geometric Approach For Trajectory Tracking Control Of Aerial Robots

Rotary-wing UAVs with the capability to hover and vertical takeoff andlanding (VTOL) are usually called aerial robots. Compared with fixed-wingUAVs, the aerial robots have many advantages in performing the task ofsearching and surveillance, especially in the complex environments. As a result,they have gained much attention in military and civilian applications in recentyears. Most flight missions require the aerial vehicles to possess the ability ofstrong maneuverability, which brings great challenge to the control system design.Conventionally, people usually utilize a group of local coordinates such asEuler angles to handle the modeling and control problems in Euclidean vectorspace. This often produces a local model and an incomplete result. The reason isthat the configuration space of an aerial robot is a Lie group, which is locallyhomeomorphic to an Euclidean space. Thus, by using only a set of coordinates,the system can’t be globally described in Euclidean vector space. To avoid thesedrawbacks, this thesis considers the intrinsic geometric characteristics in aerialrobotic dynamic modeling, and presents a general geometric framework foranalysis and trajectory tracking control system design of aerial robots.To start with, a general dynamic model of aerial robots on SE(3) isestablished. Since the potential energy of aerial robots is not left-invariant, theEuler-Poincare equation can’t be used to describe such systems. Therefore, thisthesis introduces Lagrange function and the variation on the tangent bundles ofSE(3). By using Hamiltonian principle of least action, a global geometricdescription on SE(3) for aerial robots is obtained. Based on this description, weexpress the dynamics of aerial robots by using the connection on the tangentbundles and the force and moment on the cotangent bundles. As examples, thismethod is effectively utilized to describe the modeling of two types of aerialrobots.Further, a unified mathematical description for trajectory tracking controlproblems of the full-actuated aerial robots is presented, and we generalize the PDcontrol law of the non-potential system on Riemann manifolds to this problem ina coordinate-free way. With an appropriate error function and a transportmapping, a measure of the configuration and velocity error on the tangentbundles is established, thus the trajectory tracking problem is reduced to a stabilization problem for the error system. By compensating the potential energywith a feedforward controller, the error system is converted into a non-potentialsystem, therefore the PD control is applied to the trajectory tracking control ofthe full-actuated ones. And the exponential stability of the error system’sequilibrium point is successfully proved. Numerical simulations on a non-planarhexrotor show that the controller can work effectively in trajectory tracking ofsome typical reference curves.Finally, the aforementioned control law is extended to under-actuated aerialrobots. By decomposing the trajectory tracking problem of the under-actuatedaerial robots into two full-actuated control modes, namely the position trackingmode and the attitude tracking mode, we present a PD plus feedforwardcontroller for this problem. And it is proved that the system is exponentiallystable. Numerical simulations on the quadrotor control show the feasibility ofthis approach.