Trajectory Tracking Control Of A Quadrotor With Cable-suspended Load

As a new type of air transportation,the quadrotor with cable-suspended load has a wide application prospect.Cable-suspended systems is underactuated,strongly coupling and nonlinear,the control of which is challenging.From the point of differential flatness,model building,flatness analysis,tracking control and trajectory planning for the cable-suspended system has been carried out,which has opened up new ideas for the related research.The main research contents of this paper include:1.The system is decoupled into quadrotor attitude control subsystem and cable-suspended subsystem:1)By introducing quaternions,the dynamic model of attitude motion of quadrotor is established.Then the attitude controller is deduced based on the lyapunov stability.2)The dynamics model of the cable-suspended subsystem was established based on the multi-rigid-body dynamics theory.The generalized coordinate with the same dimension of the system freedom is selected and the generalized force is calculated.Then the Lagrange dynamic model of the system is established.With fewer equations,no binding force and linear generalized acceleration,the model lays a foundation for the analysis and control of the system.2.The quadrotor-load system is proved to be differentially flat with the load position and the quadrotor yaw serving as the flatness outputs,and the analytic functions between the system states and the flat outputs are derived,so the system dynamics can be observed in the flat outputs space.Then the trajectory and control force evolution of the system are studied by simulation aiming at the circumferential motion of the load in the horizontal plane and the sinusoidal motion in the vertical plane.3.Aiming at the trajectory tracking control problem of the cable-suspended subsystem in three-dimensional space,a general algorithm for the differential flat system to be extended to linear controllable system through dynamic feedback is proposed,based on which an accurate feedback linearized trajectory tracking controller is designed.The system is transformed into a linear closed-loop system by an endogenous dynamic feedback in the 2-D plane.This process is standardized with the concept of differential geometry,After 2 dynamic expansions and variable substitutions,the original system is extended to a linear controllable system whose total relative orders are equal to the system state dimensions.Based on Hurwitz stability criterion,a dynamic feedback controller with exponential convergence of position error is designed.This method can be used as a standard method for a class of nonlinear systems.The spiral curve in 3-D space and the circular curve in the horizontal plane with varying frequency are taken as the reference trajectory for simulation.4.A trajectory planning algorithm is proposed based on the differential flatness of the system and the convex inclusion of the spline curve.The model of trajectory planning under inequality constraints was given.Based on differential flatness,the constraints of state variables and tension on cable were mapped into the flat output and its high derivatives.The feasible set is nonlinear.By use of the theory of semi-infinite optimization,a convex polytope with a given number of facets is used to replace the original nonlinear constraints by the convex polytopic approximation.The spline curve was introduced to parameterize the output curve,which is a convex linear combination of control vertexes.Then,the constraints of curve was transformed onto control vertexes.Furthermore,it was proved that the constraints of curve’s derivatives can also be transformed onto control vertexes.So the trajectory planning problem was changed into an optimization problem of searching the finite control vertexes of spline curve under linear inequality constraints,The planning algorithm was given.Simulation results verify the effectiveness of the algorithm.5.The experiment platform for transport system is designed based on open source flight controller,then the control system is verified and tested in two cases: point to point transportation and circle trajectory tracking.The experimental results show that the proposed control method has the advantages of high control accuracy and fast response when the large angle oscillation occurs on the rope.