Double-Hoist Dynamics and Oscillation Control of a Quadcopter Carrying a Swinging and Twisting Load

Quadcopters can serve as aerial cranes that are able to suspend large-size loads below the fuselage for material-handling services. Double-hoist mechanisms help quadcopter to transport bulky loads effectively. However, double-hoist mechanisms exhibit strong coupling effect among the quadcopter’s attitude, load swing, and load twisting. Different from the single-hoist mechanisms, no effects have been directed at quadcopter slung loads with double-hoist mechanisms. A novel nonlinear dynamics model of a quadcopter carrying a distributed-mass load by double-hoist mechanisms is given in this article. This explicit model captures the coupling between the quadcopter and load motion. Then a novel control method is proposed to reduce the swing and twisting of the load simultaneously. The simulation results illustrate that the control method succeeds in limiting the unwanted oscillations of quadcopter attitudes, load swing and twisting.

3 cause load swing and twisting, which result in oscillations of vehicle attitudes because of coupling impacts. However, the difficulty of accurately sensing the load states is the barrier for rejecting external disturbances toward the application of previous control methods. A new control system is designed in this article. This proposed controller is utilized to reduce the swing and mitigate the twisting of the load simultaneously. Meanwhile, the controller can also reject the external disturbances without measuring the load oscillations on-the-fly.
A primary contribution of this article is a dynamic model of aerial vehicles transporting distributed-mass loads by double-hoist mechanisms. To the best of our knowledge, it is a novel model using double-hoist dynamics of the slung-load below a quadcopter's fuselage. This model is provided in complete dynamic equations so that it can be used by other researchers. In addition, it includes the coupling among the vehicle attitude, load swing and twisting. The new control method is another contribution of this article. The control method suppresses the load swing and twisting, stabilizes the quadcopter's attitudes, rejects the external disturbances, and is easy to implement.
Simulation results demonstrate that the new controller performs well under different flight distances and different load length and mass. The modeling approach and control method are also applicable to the dynamics and control of other types of aerial cranes, such as tiltrotor or helicopter transporting large loads.
The rest of this article is organized as following. The nonlinear dynamic model of a quadcopter with a distributed-mass load is derived in Section 2. A method for controlling the quadcopter attitudes, and suppressing the suspended-load swing and twisting is proposed in Section 3.
Numerous simulations are performed to validate the disturbance rejection, effectiveness, and robustness of the proposed control method in Section 4. Finally, conclusions are drawn in Section 5. Motion of the quadcopter is divided into the displacement, x, along the Nx direction, the displacement, y, along the Ny direction, the displacement, z, along the Nz direction, yaw attitude, Another rigid link, which is parallel to the DxDy plane, connects two suspension points, P, Q, to the point, S. The distance between quadcopter center, D*, and suspension point, Q, is lADx+lBDy+lCDz, while that between the quadcopter center, D*, and suspension point, P, is -lADx-

Modeling
(1) (2) where Eq. (1) is the generalized speed of quadcopter, and Eq.     6 The angular velocity of quadcopter in the Newtonian frame N is: The velocity of quadcopter center of mass D* in the Newtonian frame N is: The angular velocity of the load is: The angular velocity of cables C1, C2 are: The derivative of Eqs. The generalized inertia forces in [33] are: where ID, IB are the inertia matrix of Quadcopter (D) and load (B), respectively. They satisfy: 7 A part of generalized active forces in [33] is due to the force of gravity on the quadcopter and load. Another part of the generalized active forces is due to the thrust force and moments produced by the rotors. Thus, the generalized active force is: where FzDz is the thrust force along the Dz direction. In order to maintain the quadcopter flight altitude, the magnitude of the thrust force along the Nz direction is FzDz· Nz for balancing gravity of the quadcopter and load.
The quadcopter, cables, and load generate a three-dimensional four-bar linkage, in which contains three velocity constraints:  sin cos cos sin sin cos cos sin cos 0 The Kane's equation in [33] describes that the sum of generalized inertia forces in Eq. (13) and generalized active forces in Eq. (15) should be limited to zero. Then the nonlinear dynamic equations of the motion in Fig. 1 yield: where M is the mass matrix, U is the column matrix of derivatives of generalized speed ur with respect to time, and Z is the column matrix of gravity terms, centrifugal and Coriolis terms, and input terms.
The dynamical model in Eq. (19) includes twelve system states, x, y, z, φ, θ, ψ, β1, γ1, β2, γ2, ε, δ, and four inputs, Fz, Mφ, Mθ, Mψ. Therefore, it is an under-actuated system that requires a sophisticated control system. The control objectives consist of three parts, i) attitude control of the quadcopter, ii) swing and twisting suppression of the distributed-mass load, and iii) externaldisturbance rejection. To achieve the objectives, a combination of feedback control and hybrid filter will be presented in this article. A feedback controller regulates the quadcopter attitude and rejects external disturbances, while a hybrid filter attenuates the load swing and twisting caused by pilot commands.

Oscillation Suppression
This section presents a combined control scheme including a feedback controller and three prefilters. A model-following controller (MFC) regulates the quadcopter's attitude by following the states of a prescribed model and attenuating the tracking errors, while three prefilters reduces both the load swing and twisting by modifying the pilot commands.
The configuration of the combined feedback and prefilter control is shown in Fig. 2. The operator generates pilot commands, ψr, θr, φr, via the remote-control transmitter. Three prefilters (hybrid filter) modify the pilot commands to produce modified commands, ψs, θs, φs. The measurement of the quadcopter attitude, ψ, θ, φ, is used to adjust the thrust moments Mψ, Mθ, Mφ, in a feedback control loop. Then, the attitudes are forced to track the modified commands by the MFC controller.
The frequencies and corresponding damping ratios are estimated by system parameters and are applied to design the three filters. The combined feedback controller and prefilter drive the aerial cranes towards the desired position with minimal oscillations of the quadcopter attitude, load swing and twisting.

MFC Controller for Attitude Regulation
The dynamics of a quadcopter slung load in Eq. (19) is too complicated to analytically design an MFC controller. Oscillations of the quadcopter and load are assumed to be small, and the model is also assumed to undergo planar motions. Then three linearized models can be derived in the where ζm is the damping ratio of the prescribed model, ωm is the frequency of the prescribed model, ψs, θs, φs, are modified commands, ψm, θm, φm, are model outputs. A prescribed model with reasonable damping ratio (0.707) and reasonable settling time (≤4s) is utilized. The desired closedloop poles were designed to be approximately -1±j1. Thus, the frequency, ωm, was 1.41 rad/s by using the pole placement method. In addition, the asymptotic tracking control law of the MFC controller is expressed as: where kψp, kψd, kθp, kθd, kφp, kφd, are control gains, and Bψ, Bθ, Bφ, are coefficients. Suitable tracking errors can be achieved by designing control gains. The coupling effect between the aerial vehicle and suspended load decreases the real part of the eigenvalues in Eq. (21) and increases the system damping. Given that the quadcopter mass, mD, and moment of inertia, Ixx, Iyy, Izz are 85 kg, 4.5 kg m 2 , 4.5 kg m 2 , 6 kg•m 2 , respectively. Thus, the prescribed eigenvalues of attitudes ψ, θ, φ, in Eq.
Resulting from the three linearized models and the MFC controller, each plane includes two linearized frequencies and corresponding damping ratios. The first-mode damping ratio is near zero, while the second-mode damping ratio is near one. The second-mode oscillations can be neglected because corresponding amplitudes may damp quickly. Actually, the three-dimensional four-bar linkage shown in Fig. 1 includes three natural frequencies with near-zero damping, and the MFC controller adds another three frequencies with high damping. Therefore, only three first-mode frequencies and damping ratios in three planar motions should be considered because they exbibit low frequency and near zero damping.
The linearized frequencies are dependent on the system parameters, such as quadcopter mass, mD, cable length, lS, load length, lL, load mass, mL, and suspension distances, lA, lB, lC.

Hybrid Filter for Suppressing Load Oscillations
The load swing and twisting caused by external disturbance might also induce oscillations of quadcopter attitude because of coupling effect between quadcopter and load. The damping effect 11 in the MFC controller can attentuate the oscillations of the quadcopter attitude. Then the load oscillations can also be reduced by MFC controller because of the coupling effect. However, the MFC controller should not be used to suppress the load swing and twisting caused by pilot commands. This is because the settling time for attentuating load oscillations by MFC controller would be very long.
In addition to regulating the quadcopter's attitude with the MFC controller, it is useful to design the prefilter to suppress load oscillations induced by pilot commands. The prefilter is a combination of discrete-and continuous-time function, which inherent in the limited motion of the quadcopter results in a limited response to oscillations of the suspended-load.
The nonlinear dynamics of the quadcopters slung loads in Eq. (19) with the MFC controller in Eqs. (20)(21) can be approximated near the equilibria as three second-order systems with linearized frequencies and damping ratios shown in Fig. 3. The response of a second-order harmonic oscillator resulting from an impulse, A1δ(n-0), at time zero and one continuous function, c(τ), is: where ωk, ζk are linearized frequency and damping ratio shown in Fig. 3. The vibrational amplitude of the response (22) is given by: The vibration amplitudes can be reduced to zero when Eqs. (24) and (25) are limited to zero. In order to ensure that the modified pilot-commands reach the same set-point as the original commands, the unit-gain constraint should be satisfied: 12 The impulse and continuous function should be positive because the negative magnitude would annoy the operator. Setting Eq. (24) and Eq. (25) to zero and solving unit-gain constraint in Eq. (26) in a time-optimal solution yields a hybrid filter, h(τ), of impulse and continuous function: where r is the modified factor of the continuous function. The magnitudes, c1, A1, are given by: The hybrid filter (27) The hybrid filter (27) is different from the input shaper (a series of impulses) and command smoother (continuous function). The duration of hybrid filter (27) can be designed by changing the modified factor, r. The modified factor, r, has a large effect on the frequency insensitivity and duration of the hybrid filter. Increasing modified factor, r, increases frequency insensitivity and duration of the hybrid function. The 5% frequency insensitivity defined in [36] for the hybrid filter (27) ranges from 0.9680 to 1.0321 when modified factor, r, is set to 0.1.
Dynamic analyses indicate that oscillations with three linearized frequencies and damping ratios in the Fig. 3 should be suppressed. Therefore, three hybrid filters with three linearized frequencies and damping ratios are placed in series to attenuate swing and twisting of the load.
The MFC controller shown in Eqs. (20) and (21) regulates the quadcopter's attitudes and rejects external disturbances, and three hybrid filters shown in Eq. (27) attenuates the load oscillations caused by pilot commands.
By double-hoist mechanisms and two suspension links shown in Fig. 1, the swing angles, β1, γ1, β2, γ2, the slope angle, δ, and the twist angle, ε, are fully coupled with yaw attitude, ψ, pitch 13 attitude, θ, and the roll attitude, φ. The coupling effects benefit disturbance rejection. This is because the MFC controller stabilizes the attitudes, and will also reduces the load oscillations by the coupling impact. Therefore, the load oscillations caused by external disturbances, which do not need to measure on-the-fly, can be suppressed by using the double-hoist mechanisms and two suspension links. Single-hoist configuration and cable connection directly to aerial vehicles cannot cause fully coupling effect between the load oscillations and quadcopter attitudes.

Numerical Verification
In

Disturbance Rejection
External disturbances might cause load oscillations in many cases. The first group of simulations is conducted to investigate the control performance of rejecting external disturbances.  swing and slope, γ1, γ2, δ; c) load swing and twisting, β1, β2, ε.
in Fig. 1 create fully coupling effect between the quadcopter and load. The presented controller reduced oscillations of quadcopter and load caused by external disturbances using the coupling impact.     6b shows the residual amplitude and settling time of the load swing, γ1, β1. The simulated results of the load swing, γ1, are similar to that of the quadcopter attitude, φ, observed in Fig. 6a.

Effectiveness of Oscillation Suppression
Moreover, the frequency, γ1, is also restricted by the frequency linked to the value in Fig. 3. The results for load twisting, ε, and slope, δ, are shown in Fig. 6c. The dynamics of the load slope are also similar to the that of the quadcopter attitude, φ, and the load swing, γ1. The residual amplitude and settling time of the load twisting changed slightly for load length shown. The combined MFC controller and three prefilters limited the residual amplitude of the load twisting and slope to below 0.01 º and 0.1 º for all the cases shown in Fig. 6c. The combined control method attenuated an average of 99.9 % and 97.7 % more residual amplitude of the load twisting and slope than the MFC controller and suppresses total settling time.

Robustness to Modeling Errors in System Parameter
An additional set of simulations was performed to assess the robustness of the combined control scheme for various modeling errors in the system parameters.    Meanwhile, the settling time of the load swing, β1, keeps zero until 3.5 m, and then increases slowly.
The combined MFC controller and three prefilters limited the residual deflection and settling time of the load swing, γ1, β1, to under 0.2 º , 0.0 s, 0.1 º , 0.0 s, respectively, for the parameter ranges shown.
The simulated load twisting and slope are shown in Fig. 7c. When using the MFC controller, the residual amplitude of the load twisting changes slightly and the settling time of the load twisting decreases with increasing load length. The residual amplitude of load slope contains a peak as the load length is varied. The settling time of load slope decreases with increasing load length.
Therefore, the combined control scheme reduced residual amplitude and settling time of the load twisting over the MFC controller by an average of 98.0% and 100%, and those of the load slope by an average of 97.6% and 100%.
Figures. 4-7 demonstrates that the combined MFC controller and three prefilters is beneficial for reducing the oscillations of the quadcopter's attitude, load swing, and load twisting. Therefore, the combined control method provides a safer environment to operate the quadcopter slung load, and a higher efficiency for transferring loads to the desired position precisely.

Conclusions
The dynamic effects and oscillation suppression of quadcopters transporting large-size loads were presented in this article. A nonlinear dynamic model of a quadcopter carrying a distributedmass slender load with double-hoist dynamics was derived using Kane's method. The model includes the motions of the quadcopter's attitude, load swing, and load twisting. A combined MFC controller and three prefilters was presented for controlling the quadcopter's attitude, load swing, and load twisting. The MFC controller adjusts the quadcopter's attitudes and rejects external disturbances, while the hybrid filter reduces the load swing and twisting caused by pilot commands.
Numerous simulations verify the effectiveness and robustness of the combined control scheme for various flight distances and system parameters.
This article only provides numerical verification. Because the prefilter is difficult to achieve good experimental results on small-scale aerial vehicles in the laboratory, the future study would be experimental verification of theoretical findings on the large-scale quadcopter (dozens of kilograms).