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Research Papers

Stabilization of a Rigid Body Payload With Multiple Cooperative Quadrotors OPEN ACCESS

[+] Author and Article Information
Farhad A. Goodarzi

Department of Mechanical and
Aerospace Engineering,
George Washington University,
Washington, DC 20052
e-mail: fgoodarzi@gwu.edu

Taeyoung Lee

Associate Professor
Department of Mechanical and
Aerospace Engineering,
George Washington University,
Washington, DC 20052
e-mail: tylee@gwu.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received November 6, 2015; final manuscript received June 7, 2016; published online August 10, 2016. Assoc. Editor: Yongchun Fang.

J. Dyn. Sys., Meas., Control 138(12), 121001 (Aug 10, 2016) (17 pages) Paper No: DS-15-1557; doi: 10.1115/1.4033945 History: Received November 06, 2015; Revised June 07, 2016

This paper presents the full dynamics and control of arbitrary number of quadrotor unmanned aerial vehicles (UAVs) transporting a rigid body. The rigid body is connected to the quadrotors via flexible cables where each flexible cable is modeled as a system of arbitrary number of serially connected links. It is shown that a coordinate-free form of equations of motion can be derived for the complete model without any simplicity assumptions that commonly appear in other literature, according to Lagrangian mechanics on a manifold. A geometric nonlinear controller is presented to transport the rigid body to a fixed desired position while aligning all of the links along the vertical direction. A rigorous mathematical stability proof is given and the desirable features of the proposed controller are illustrated by numerical examples and experimental results.

FIGURES IN THIS ARTICLE
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Quadrotor UAVs are being considered for various missions such as Mars surface exploration, search and rescue, and particularly payload transportation. There are various applications for aerial load transportation such as usage in construction, military operations, emergency response, or delivering packages. Load transportation with UAVs can be performed using a cable or by grasping the payload [1,2]. There are several limitations for grasping a payload with UAVs such as in situations where the landing area is inaccessible, or, it transporting a heavy/bulky object by multiple quadrotors (Fig. 1).

Load transportation with the cable-suspended load has been studied traditionally for a helicopter [3,4] or for small UAVs such as quadrotor UAVs [57].

In most of the prior works, the dynamics of aerial transportation has been simplified due to the inherent dynamic complexities. For example, it is assumed that the dynamics of the payload is considered completely decoupled from quadrotors, and the effects of the payload and the cable are regarded as arbitrary external forces and moments exerted to the quadrotors [810], thereby making it challenging to suppress the swinging motion of the payload actively, particularly for agile aerial transportations.

Recently, the coupled dynamics of the payload or cable has been explicitly incorporated into control system design [11]. In particular, a complete model of a quadrotor transporting a payload modeled as a point mass, connected via a flexible cable is presented, where the cable is modeled as serially connected links to represent the deformation of the cable [12,13]. In these studies, the payload simplified and considered as a point mass without the attitude and the moment of inertia. In another study, multiple quadrotors transporting a rigid body payload have been studied [14], but it is assumed that the cables connecting the rigid body payload and quadrotors are always taut. These assumptions and simplifications in the dynamics of the system reduce the stability of the controlled system, particularly in rapid and aggressive load transportation where the motion of the cable and payload is excited nontrivially.

The other critical issue in designing controllers for quadrotors is that they are mostly based on local coordinates. Some aggressive maneuvers are demonstrated at Ref. [15] based on Euler angles. However, they involve complicated expressions for trigonometric functions, and they exhibit singularities in representing quadrotor attitudes, thereby restricting their ability to achieve complex rotational maneuvers significantly. A quaternion-based feedback controller for attitude stabilization was shown in Ref. [16]. By considering the Coriolis and gyroscopic torques explicitly, this controller guarantees exponential stability. Quaternions do not have singularities but, as the three-sphere double-covers the special orthogonal group, one attitude may be represented by two antipodal points on the three-sphere. This ambiguity should be carefully resolved in quaternion-based attitude control systems, otherwise they may exhibit unwinding, where a rigid body unnecessarily rotates through a large angle even if the initial attitude error is small [17]. To avoid these, an additional mechanism to lift attitude onto the unit-quaternion space is introduced [18].

Recently, the dynamics of a quadrotor UAV is globally expressed on the special Euclidean group, SE(3), and nonlinear control systems are developed to track outputs of several flight modes [19]. Several aggressive maneuvers of a quadrotor UAV are demonstrated based on a hybrid control architecture, and a nonlinear robust control system is also considered in Refs. [20,21]. As they are directly developed on the special orthogonal group, complexities, singularities, and ambiguities associated with minimal attitude representations or quaternions are completely avoided. The proposed control system is particularly useful for rapid and safe payload transportation in complex terrain, where the position of the payload should be controlled concurrently while suppressing the deformation of the cables.

Comparing with the prior work of the authors in Refs. [2224] and other existing studies, this paper is the first study considering a complete model which includes a rigid body payload with attitude, arbitrary number of quadrotors, and flexible cables. Also, this study presents a control system to stabilize the rigid body at desired position. Geometric nonlinear controllers are developed for the presented model. More explicitly, we show that the rigid body payload is asymptotically transported into a desired location, while aligning all of the links along the vertical direction corresponding to a hanging equilibrium.

In short, new contributions and the unique features of the dynamics model and control system proposed in this paper compared with other studies are as follows: (i) it is developed for the full dynamic model of arbitrary number of multiple quadrotor UAVs on SE(3) transporting a rigid body connected via flexible cables, including the coupling effects between the translational dynamics and the rotational dynamics on a nonlinear manifold, (ii) the control systems are developed directly on the nonlinear configuration manifold in a coordinate-free fashion. Thus, singularities of local parameterization are completely avoided to generate agile maneuvers in a uniform way, (iii) a rigorous Lyapunov analysis is presented to establish stability properties without any timescale separation assumption or singular perturbation, (iv) an integral control term is proposed to guarantee asymptotical convergence of tracking error variables in the presence of unstructured uncertainties in both rotational and translational dynamics, and (v) the proposed algorithm is validated with experiments for payload transportation with multiple cooperative quadrotor UAVs. A rigorous and complete mathematical analysis for multiple quadrotor UAVs transporting a payload on SE(3) with experimental validations for payload transportation maneuvers is unprecedented.

This paper is organized as follows. A dynamic model is presented and the problem is formulated at Sec. 2. Control systems are constructed at Secs. 3 and 4, which are followed by numerical examples in Sec. 5. Finally, experimental results are presented in Sec. 6.

Consider a rigid body with the mass m0 and the moment of inertia J03×3, being transported with n arbitrary number of quadrotors as shown in Fig. 2. The location of the mass center of the rigid body is denoted by x03, and its attitude is given by R0SO(3), where the special orthogonal group is given by SO(3)={R3×3RTR=I,det(R)=1}. We choose an inertial frame {e1,e2,e3} and body-fixed frame {b1,b2,b3} attached to the payload. We also consider a body-fixed frame attached to the ith quadrotor {b1i,b2i,b3i}. In the inertial frame, the third axes e3 points downward along the gravity and the other axes are chosen to form an orthonormal frame.

The mass and the symmetric moment of inertia of the ith quadrotor are denoted by mi and Ji3×3, respectively. The cable connecting each quadrotor to the rigid body is modeled as an arbitrary numbers of links for each quadrotor with varying masses, mij, and lengths, lij. The direction of the jth link of the ith quadrotor, measured outward from the quadrotor toward the payload is defined by the unit vector qijS2, where S2={q3|q=1}, where the mass and length of that link is denoted with mij and lij, respectively. The number of links in the cable connected to the ith quadrotor is defined as ni. The configuration manifold for this system is given by SO(3)×3×(SO(3)n)×(S2)Σi=1nni.

The ith quadrotor can generate a thrust force of fiRie33 with respect to the inertial frame, where fi is the total thrust magnitude of the ith quadrotor. It also generates a moment Mi3 with respect to its body-fixed frame. Also, we define Δxi and ΔRi3 as fixed disturbances applied to the ith quadrotor's translational and rotational dynamics, respectively. It is also assumed that an upper bound of the infinite norm of the uncertainty is known Display Formula

(1)Δxδ

for a positive constant δ.

Throughout this paper, λm(A) and λM(A) denote the minimum eigenvalue and the maximum eigenvalue of a square matrix A, respectively, and λm and λM are the minimum eigenvalue and the maximum eigenvalue of the inertia matrix J, i.e., λm=λm(J) and λM=λM(J). The two-norm of a matrix A is denoted by A. The standard dot product is denoted by x·y=xTy for any x,y3.

Lagrangian.

The kinematics equations for the links, payload, and quadrotors are given by Display Formula

(2)q˙ij=ωij×qij=ω̂ijqij
Display Formula
(3)R˙0=R0Ω̂0
Display Formula
(4)R˙i=RiΩ̂i

where ωij3 is the angular velocity of the jth link in the ith cable satisfying qij·ωij=0. Also, Ω03 is the angular velocity of the payload and Ωi3 is the angular velocity of the ith quadrotor, expressed with respect to the corresponding body-fixed frame. The hat map·̂:3SO(3) is defined by the condition that x̂y=x×y for all x,y3. More explicitly, for a vector a=[a1,a2,a3]T3, the matrix â is given by Display Formula

(5)â=[0a3a2a30a1a2a10]

This identifies the Lie algebra SO(3) with 3 using the vector cross product in 3. The inverse of the hat map is denoted by the vee map, :SO(3)3.

The position of the ith quadrotor is given by Display Formula

(6)xi=x0+R0ρia=1niliaqia

where ρi3 is the vector from the center of mass of the rigid body to the point that ith cable is connected to the rigid body. Similarly, the position of the jth link in the cable connecting the ith quadrotor to the rigid body is given by Display Formula

(7)xij=x0+R0ρia=j+1niliaqia

We derive equations of motion according to Lagrangian mechanics. Total kinetic energy of the system is given by Display Formula

(8)T=12m0x˙02+i=1nj=1ni12mijx˙ij2+12i=1nmix˙i2+12i=1nΩi·JiΩi+12Ω0·J0Ω0

The gravitational potential energy is given by Display Formula

(9)V=m0ge3·x0i=1nmige3·xii=1nj=1nimijge3·xij

where it is assumed that the unit vector e3 points downward along the gravitational acceleration as shown at Fig. 2. The corresponding Lagrangian of the system is L=TV.

Euler–Lagrange Equations.

Coordinate-free form of Lagrangian mechanics on the two-sphere S2 and the special orthogonal group SO(3) for various multibody systems has been studied in Refs. [25,26]. The key idea is representing the infinitesimal variation of RiSO(3) in terms of the exponential map Display Formula

(10)δRi=ddε|ε=0Riexp(εη̂i)=Riη̂i

for ηi3. The corresponding variation of the angular velocity is given by δΩi=η˙i+Ωi×ηi. Similarly, the infinitesimal variation of qijS2 is given by Display Formula

(11)δqij=ξij×qij

for ξij3 satisfying ξij·qij=0. Using these, we obtain the following Euler–Lagrange equations.

Proposition 1. The equations of motion for the proposed payload transportation system are as follows:Display Formula

(12)MTx¨0i=1nj=1niM0ijlijq¨iji=1nMiTR0ρ̂iΩ˙0=MTge3+i=1n(fiRie3+Δxi)i=1nMiTR0Ω̂02ρi
Display Formula
(13)J¯0Ω˙0+i=1nMiTρ̂iR0Tx¨0i=1nj=1niM0ijlijρ̂iR0Tq¨ij=i=1nρ̂iR0T(fiRie3+MiTge3+Δxi)Ω̂0J¯0Ω0
Display Formula
(14)k=1niM0ijlikq̂ij2q¨ikM0ijq̂ij2x¨0+M0ijq̂ij2R0ρ̂iΩ˙0=M0ijq̂ij2R0Ω̂02ρiq̂ij2(M0ijge3fiRie3+Δxi)
Display Formula
(15)JiΩi+Ωi×JiΩi=Mi+ΔRi

Here, the total mass MT of the system and the mass of the ith quadrotor and its flexible cable MiT are defined asDisplay Formula

(16)MT=m0+i=1nMiT,MiT=j=1nimij+mi

and the constants related to the mass of links are given asDisplay Formula

(17)M0ij=mi+a=1j1mia

The equations of motion can be rearranged in a matrix form as follows:Display Formula

(18)NX˙=P

where the state vectorXDX withDX=6+3i=1nni is given byDisplay Formula

(19)X=[x˙0,Ω0,q˙1j,q˙2j,,q˙nj]T

and matrixNDX×DX is defined asDisplay Formula

(20)N=[MTI3Nx0Ω0Nx01Nx02Nx0nNΩ0x0J¯0NΩ01NΩ02NΩ0nN1x0N1Ω0Nqq100N2x0N2Ω00Nqq20Nnx0NnΩ000Nqqn]

where the submatrices are defined asDisplay Formula

(21)Nx0Ω0=i=1nMiTR0ρ̂i;NΩ0x0=Nx0Ω0TNx0i=[M0i1li1I3,M0i2li2I3,,M0iniliniI3]NΩ0i=[M0i1li1ρ̂iR0T,M0i2li2ρ̂iR0T,,M0iniliniρ̂iR0T]Nix0=[M0i1q̂i12,M0i2q̂i22,,M0iniq̂ini2]TNiΩ0=[M0i1q̂i12R0ρ̂i,M0i2q̂i22R0ρ̂i,,M0iniq̂ini2R0ρ̂i]T

and the submatrixNqqi3ni×3ni is given byDisplay Formula

(22)Nqqi=[M011li1I3M012li2q̂i22M01niliniq̂ini2M021li1q̂i12M022li2I3M02niliniq̂ini2M0ni1li1q̂i12M0ni2li2q̂i22M0niniliniI3]

ThePDX matrix isDisplay Formula

(23)P=[Px0,PΩ0,P1j,P2j,,Pnj]T

and submatrices ofP matrix are also defined as

Px0=MTge3+i=1n(fiRie3+Δxi)i=1nMiTR0Ω̂02ρiPΩ0=Ω̂0J¯0Ω0+i=1nρ̂iR0T(MiTge3fiRie3+Δxi)Pij=q̂ij2(fiRie3+M0ijge3+Δxi)+M0ijq̂ij2R0Ω̂02ρi+M0ijlijq˙ij2qij

Proof. See Appendix A.

These equations are derived directly on a nonlinear manifold. The dynamics of the payload, flexible cables, and quadrotors are considered explicitly, and they avoid singularities and complexities associated to local coordinates.

Control Problem Formulation.

Let x0d3 be the desired position of the payload. The desired attitude of the payload is considered as R0d=I3×3, and the desired direction of links is aligned along the vertical direction. The corresponding location of the ith quadrotor at this desired configuration is given by Display Formula

(24)xid=x0d+ρia=1niliae3

We wish to design control forces fi and control moments Mi of quadrotors such that this desired configuration becomes asymptotically stable.

Simplified Dynamic Model.

Control forces for each quadrotor are given by fiRie3 for the given equations of motion (12)(15). As such, the quadrotor dynamics is underactuated. The total thrust magnitude of each quadrotor can be arbitrary chosen, but the direction of the thrust vector is always along the third body-fixed axis, represented by Rie3. But, the rotational attitude dynamics of the quadrotors are fully actuated, and they are not affected by the translational dynamics of the quadrotors or the dynamics of links.

Based on these observations, in this section, we simplify the model by replacing the fiRie3 term by a fictitious control input ui3, and design an expression for ui to asymptotically stabilize the desired equilibrium. In other words, we assume that the attitude of the quadrotor can be instantaneously changed. Also, Δxi are ignored in the simplified dynamic model. The effects of the attitude dynamics are incorporated in Sec. 4.

Linear Control System.

The control system for the simplified dynamic model is developed based on the linearized equations of motion. At the desired equilibrium, the position and the attitude of the payload are given by x0d and R0d=I3, respectively. Also, we have qijd=e3 and Rid=I3. In this equilibrium configuration, the control input for the ith quadrotor is Display Formula

(25)uid=fidRide3

where the total thrust is fid=(MiT+(m0/n))g.

The variation of x0 is given by Display Formula

(26)δx0=x0x0d

and the variation of the attitude of the payload is defined as

δR0=R0dη̂0=η̂0

for η03. The variation of qij can be written as Display Formula

(27)δqij=ξij×e3

where ξij3 with ξij·e3=0. The variation of ωij is given by δωij3 with δωij·e3=0. Therefore, the third element of each of ξij and δωij for any equilibrium configuration is zero, and they are omitted in the following linearized equations. The state vector of the linearized equation is composed of CTξij2, where C=[e1,e2]3×2. The variation of the control input δui3×1, is given as δui=uiuid.

Proposition 2. The linearized equations of the simplified dynamic model are given byDisplay Formula

(28)Mx¨+Gx=Bδu+g(x,x˙)

whereg(x,x˙) corresponds to the higher-order term and the state vectorxDx withDx=6+2i=1nni is given by

x=[δx0,η0,CTξ1j,CTξ2j,,CTξnj]

andδu=[δu1T,δu2T,,δunT]T3n×1. The matrixMDx×Dx are defined as

M=[MTI3Mx0Ω0Mx01Mx02Mx0nMΩ0x0J¯0MΩ01MΩ02MΩ0nM1x0M1Ω0Mqq100M2x0M2Ω00Mqq20Mnx0MnΩ000Mqqn]

where the submatrices are defined asDisplay Formula

(29)Mx0Ω0=i=1nMiTρ̂i;MΩ0x0=Mx0Ω0TMx0i=[M0i1li1ê3C,M0i2li2ê3C,,M0iniliniê3C]MΩ0i=[M0i1li1ρ̂iC,M0i2li2ρ̂iC,,M0iniliniρ̂iC]Mix0=[M0i1CTê3,M0i2CTê3,,M0iniCTê3]
Display Formula
(30)MiΩ0=[M0i1CTê3ρ̂i,M0i2CTê3ρ̂i,,M0iniCTê3ρ̂i]

and the submatrixMqqi2ni×2ni is given byDisplay Formula

(31)Mqqi=[Mi11li1I2Mi12li2I2Mi1niliniI2Mi21li1I2Mi22li2I2Mi2niliniI2Mini1li1I2Mini2li2I2MininiliniI2]

The matrixGDx×Dx is defined as

G=[0000000GΩ0Ω0000000G1000000G20000000Gn]

whereGΩ0Ω0=i=1n(m0/n)gρ̂iê3 and the submatricesGi2ni×2ni are

Gi=diag[(MiTm0n+M0ij)ge3I2]

The matrixBDx×3n is given by

B=[I3I3I3ρ̂1ρ̂2ρ̂nBB0000BB00000BB]

whereBB=[CTê3,CTê3,,CTê3]T.

Proof. See Appendix B.

We present the following proportional-derivative-type control system for the linearized dynamics: Display Formula

(32)δui=KxixKx˙ix˙

for controller gains Kxi,Kx˙i3×Dx. Provided that Eq. (28) is controllable, we can choose the combined controller gains Kx=[Kx1T,KxnT]T,Kx˙=[Kx˙1T,Kx˙nT]T3n×Dx such that the equilibrium is asymptotically stable for the linearized equations [27]. Then, the equilibrium becomes asymptotically stable for the nonlinear Euler–Lagrange equation. The controlled linearized system can be written as Display Formula

(33)z˙1=Az1+Bg(x,x˙)

where z1=[x,x˙]T2Dx and Display Formula

(34)A=[0IM1(G+BKx)M1BKx˙],B=[0M1]

We choose Kx and Kx˙ such that A2Dx×2Dx is Hurwitz. Then for any positive definite matrix Q2Dx×2Dx, there exist a positive definite and symmetric matrix P2Dx×2Dx such that ATP+PA=Q according to Ref. [27, Theorem 3.6].

The control system designed at Sec. 3 is based on a simplifying assumption that each quadrotor can generate a thrust along any direction. In the full dynamic model, the direction of the thrust for each quadrotor is parallel to its third body-fixed axis always. In this section, the attitude of each quadrotor is controlled such that the third body-fixed axis becomes parallel to the direction of the ideal control force designed in Sec. 3. Also in the full dynamics model, we consider the Δxi in the control design and introduce a new integral term to eliminate the disturbances and uncertainties. The central idea is that the attitude Ri of the quadrotor is controlled such that its total thrust direction Rie3, corresponding to the third body-fixed axis, asymptotically follows the direction of the fictitious control input ui. By choosing the total thrust magnitude properly, we can guarantee local asymptotical stability for the full dynamic model.

Let Ai3 be the ideal total thrust of the ith quadrotor that locally asymptotically stabilize the desired equilibrium, defined as Display Formula

(35)Ai=uid+δui=KxixKx˙ix˙Kzsatσ(ex)+uid

where fid and uid3 are the total thrust and control input of each quadrotor at its equilibrium, respectively, and the following integral term exDx is added to eliminate the effect of disturbance Δxi in the full dynamic model: Display Formula

(36)ex=0t(PB)Tz1(τ)dτ

where Kz=[kzI3,kzI3,kz1I3×2,kznI3×2]3×Dx is an integral gain. For a positive constant σ, a saturation function satσ:[σ,σ] is introduced as

satσ(y)={σify>σyifσyσσify<σ

If the input is a vector yn, then the above saturation function is applied element by element to define a saturation function satσ(y):n[σ,σ]n for a vector.

From the desired direction of the third body-fixed axis of the ith quadrotor, namely, b3ciS2, is given by Display Formula

(37)b3ci=AiAi

This provides a two-dimensional constraint on the three-dimensional desired attitude of each quadrotor, such that there remains one degree-of-freedom. To resolve it, the desired direction of the first body-fixed axis b1i(t)S2 is introduced as a smooth function of time. Due to the fact that the first body-fixed axis is normal to the third body-fixed axis, it is impossible to follow an arbitrary command b1i(t) exactly. Instead, its projection onto the plane normal to b3ci is followed, and the desired direction of the second body-fixed axis is chosen to constitute an orthonormal frame [23]. More explicitly, the desired attitude of the ith quadrotor is given by Display Formula

(38)Ric=[(b̂3ci)2b1i(b̂3ci)2b1ib̂3cib1ib̂3cib1ib3ci]

which is guaranteed to be an element of SO(3). The desired angular velocity is obtained from the attitude kinematics equation, Ωic=(RicTR˙ic)3.

Define the tracking error vectors for the attitude and the angular velocity of the ith quadrotor as Display Formula

(39)eRi=12(RicTRiRiTRic),eΩi=ΩiRiTRicΩic

and a configuration error function on SO(3) as follows: Display Formula

(40)Ψi=12tr[IRicTRi]

The thrust magnitude is chosen as the length of ui, projected on to Rie3, and the control moment is chosen as a tracking controller on SO(3)Display Formula

(41)fi=Ai·Rie3
Display Formula
(42)Mi=kRieRikΩieΩikIeIi+(RiTRicΩic)JiRiTRicΩic+JiRiTRicΩ˙ic

where kRi,kΩi, and kI are positive constants and the following integral term is introduced to eliminate the effect of fixed disturbance ΔRi: Display Formula

(43)eIi=0teΩi(τ)+c2eRi(τ)dτ

where c2 is a positive constant. Stability of the corresponding controlled systems for the full dynamic model can be studied by showing the error due to the discrepancy between the desired direction b3ci and the actual direction Rie3.

Proposition 3. Consider control inputs fi, Mi defined in Eqs.(41) and(42). There exist controller parameters and gains such that, (i) the zero equilibrium of tracking error is stable in the sense of Lyapunov; (ii) the tracking errorseRi,eΩi,x,x˙ asymptotically converge to zero as t → ∞; and (iii) the integral termseIi andex are uniformly bounded.

Proof. See Appendix C.

By utilizing geometric control systems for quadrotor, we show that the hanging equilibrium of the links can be asymptotically stabilized while translating the payload to a desired position and attitude. The control systems proposed explicitly consider the coupling effects between the cable/load dynamics and the quadrotor dynamics. We presented a rigorous Lyapunov stability analysis to establish stability properties without any timescale separation assumptions or singular perturbation, and a new nonlinear integral control term is designed to guarantee robustness against unstructured uncertainties in both rotational and translational dynamics.

We demonstrate the desirable properties of the proposed control system with numerical examples. Two cases are presented. At the first case, a payload is transported to a desired position from the ground. The second case considers stabilization of a payload with large initial attitude errors.

Stabilization of the Rigid Body.

Consider four quadrotors (n = 4) connected via flexible cables to a rigid body payload. Initial conditions are chosen as

x0(0)=[1.0,4.8,0.0]Tm,v0(0)=03×1qij(0)=e3,ωij(0)=03×1,Ri(0)=I3×3,Ωi(0)=03×1R0(0)=I3×3,Ω0=03×1

The desired position of the payload is chosen as Display Formula

(44)x0d(t)=[0.44,0.78,0.5]Tm

The mass properties of quadrotors are chosen as Display Formula

(45)mi=0.755kgJi=diag[0.557,0.557,1.05]×102kg/m2

and the attitude control gains of each quadrotor are defined with the following expressions:

kRi=ωnR2JikΩi=2ωnRζRJi

where considering ωnR=8 and ζR=0.7, the attitude control gains are given by kRi=0.6724 and kΩi=0.1177 for all quadrotors and kI=kz=0.01. The linear controller gains which are Kxi and Kx˙i are calculated using the linear quadratic regulator method which each has a dimension of 12 × 46. The payload is a box with mass m0=0.5kg, and its length, width, and height are 0.6, 0.8, and 0.2 m, respectively. Each cable connecting the rigid body to the ith quadrotor is considered to be ni = 5 rigid links. All the links have the same mass of mij = 0.01 kg and length of lij = 0.15 m. Each cable is attached to the following points of the payload:

ρ1=[0.3,0.4,0.1]Tm,ρ2=[0.3,0.4,0.1]Tmρ3=[0.3,0.4,0.1]Tm,ρ4=[0.3,0.4,0.1]Tm

Numerical simulation results are presented at Figs. 3 and 4, which shows the position and velocity of the payload and its tracking errors. We have also presented the link direction and link angular velocity errors defined as

Display Formula

(46)eq=i=1mj=1niqije3
Display Formula
(47)eω=i=1mj=1niωij

Payload Stabilization With Large Initial Attitude Errors.

In the second case, we consider large initial errors for the attitude of the payload and quadrotors. Initially, the rigid body is tilted about its b1 axis by 30 deg, and the initial direction of the links is chosen such that two cables are curved along the horizontal direction. The initial conditions are given by

x0(0)=[2.4,0.8,1.0]T,v0(0)=03×1ωij(0)=03×1,Ωi(0)=03×1R0(0)=Rx(30deg),Ω0=03×1

where Rx(30 deg) denotes the rotation about the first axis by 30 deg. The initial attitude of quadrotors is chosen as

R1(0)=Ry(35deg),R2(0)=I3×3R3(0)=Ry(35deg),R4(0)=I3×3

The mass properties of quadrotors and cables are chosen as previous example. The payload mass is m = 1.0 kg, and its length, width, and height are 1.0, 1.2, and 0.2 m, respectively. Each cable is attached to the following points of the payload:

ρ1=[0.5,0.6,0.1]Tm,ρ2=[0.5,0.6,0.1]Tmρ3=[0.5,0.6,0.1]Tm,ρ4=[0.6,0.6,0.1]Tm

Figure 5 illustrates the tracking errors, and the total thrust of each quadrotor. Snapshots of the controlled maneuvers are also illustrated at Fig. 6. It is shown that the proposed controller is able to stabilize the payload and cables at their desired configuration even from the large initial attitude errors.