A trajectory tracking control law for a quadrotor with slung load

Abstract

We present a trajectory tracking controller for the full dynamics of a quadrotor vehicle carrying a slung load attached by a string. The full dynamic system is modeled as two connected subsystems, the string–load subsystem, with dynamics identical to that of a standard quadrotor in free flight, and the quadrotor subsystem with attitude kinematics and dynamics. A trajectory tracking controller for the position of the point-mass load is designed based on existing Lyapunov-based trajectory tracking controllers for free flying quadrotors which are further backstepped through the quadrotor attitude dynamics. A parameterized Lyapunov function is provided for the full system dynamics with a negative semi-definite time derivative. The proposed controller is proven to drive the load position error to zero and the origin of the error system is exponentially stable. Simulation results attest the performance of the proposed controller for aggressive trajectories and its robustness and validity are further highlighted by experimental results with a model-scale vehicle and slung load.

Introduction

The last decade oversaw the first steps and boom of research of slung load transportation with model-scale autonomous rotorcraft. Work on cranes and full size helicopters carrying slung loads provided the inspiration for the original works using model-scale rotorcraft (Bernard & Kondak, 2009). These first attempts were based on inner–outer loop designs and linearization of the vehicle, string and load systems. The load was modeled as a mass point and the control solution involved estimation of string tension forces, based on a flexible string dynamic model obtained using computer software. A comprehensive dynamic model that allows for slack strings and aerodynamic drag on the load was developed in Bisgaard, Bendtsen, and Cour-Harbo (2009).

The use of multiple small-scale aerial vehicles for transport of a slung rigid body was pioneered in Michael, Fink, and Kumar (2011) and the problem from a robotics inspired perspective where the inputs are the quadrotor positions and the system acts effectively as a parallel manipulator.

Other early works (Faust et al., 2013, Palunko et al., 2012) focused on trajectory generation for a slung load system resulting in optimal swing-free trajectories. The resulting trajectory for the quadrotor vehicle was then tracked by the vehicle but no feedback from the load position or string angle was used to improve control performance and the control derivation relies on linearization.

More recently, approaches rooted in geometric control, starting with Sreenath, Lee and Kumar (2013b) and Sreenath, Michael and Kumar (2013a), have developed the complete system dynamics and controllers directly on the nonlinear configuration manifold in a coordinate-free form. Moreover, these works consider explicitly the coupling between load and quadrotor dynamics. The resulting controllers are thus almost-global and the load is able to undergo large swings while under feedback control. While Sreenath, Michael et al. (2013a) restricted itself to the planar case, experimental results for the geometric controller (Sreenath, Lee et al., 2013b) were later presented in Tang and Kumar (2015), which focuses on trajectory generation while relaxing the condition that the string be always taut.

Typically, coordinate-free methods can be applied to obtain almost-global controller for a vehicle transporting a payload whereas when dealing with flexible strings, due to the infinite dimensionality of the problem, one cannot escape the use of approximations and linearization methods. It is also common to simplify the control problem by considering that the inner-loop corresponding to the quadrotor orientation is attained instantly, or at least in finite time, thereby suppressing the coupling of the quadrotor dynamics on the string and load subsystems.

In this brief we propose a novel look at the slung load system. By drawing a parallel between the string–load subsystem, driven by a fully-actuated vehicle, and a standard quadrotor model in free flight, existing position tracking controllers for quadrotors can be adapted to slung load position tracking. Following the proposed control design, a Lyapunov function can be constructed for the overall closed-loop system resulting in a position error system proven to be almost-globally asymptotically stable and locally exponentially stable. Since no conservatism is introduced, the proposed method has potentially better performance and convergence regions than perturbation theory or inner–outer loop based controllers.