\tensf Post Doctoral Fellow
George Washington University
Washington, DC 20052
Email: fgoodarzi@gwu.edu
Taeyoung Lee
\tensfAssociate Professor
George Washington University
Washington, DC 20052
Email: tylee@gwu.edu
###### Abstract

This paper presents the full dynamics and control of arbitrary number of quadrotor unmanned aerial vehicles (UAV) 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.

\@nthm

definitionDefinition \@nthmlemLemma \@nthmpropProposition \@nthmremarkRemark

## Nomenclature

Number of links in the -th cable

Inertial frame

Body-fixed frame of the -th quadrotor

Inertia matrix of the -th quadrotor

Angular velocity of the -th quadrotor

Gravitational acceleration

Special Orthogonal group

Special Euclidean group

## 1 Introduction

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 [?, ?]. 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.

Load transportation with the cable-suspended load has been studied traditionally for a helicopter [?, ?] or for small unmanned aerial vehicles such as quadrotor UAVs [?, ?, ?].

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 [?, ?, ?], thereby making it challenging to suppress the swinging motion of the payload actively, particularly for agile aerial transportations.

The other critical issue in designing controllers for quadrotors is that they are mostly based on local coordinates. Some aggressive maneuvers are demonstrated at [?] 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 [?]. 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 [?]. To avoid these, an additional mechanism to lift attitude onto the unit-quaternion space is introduced [?].

Recently, the dynamics of a quadrotor UAV is globally expressed on the special Euclidean group, , and nonlinear control systems are developed to track outputs of several flight modes [?]. There are also several studies using the estimations for dynamical objects developed on the special Euclidean group [?, ?]. 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 [?, ?]. As they are directly developed on the special Euclidean/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 [?, ?, ?] 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. A rigorous mathematical stability analysis is presented, and numerical and experimental validations in presence of uncertainties and disturbances are provided. More explicitly, we present the complete dynamic model of an arbitrary number of quadrotors transporting a rigid body where each quadrotor is connected to the rigid body via a flexible cable. Each flexible cable is modeled as an arbitrary number of serially connected links, and it is valid for various masses and lengths. A coordinate free form of equations of motion is derived according to Lagrange mechanics on a nonlinear manifold for the full dynamic model. These sets of equations of motion are presented in a complete and organized manner without any restrictive assumption or simplification.

Another contribution of this study is designing a control system to stabilize the rigid body at desired position. Geometric nonlinear controllers presented and generalized 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. This paper presents a rigorous Lyapunov stability analysis for the proposed controller to establish stability properties without any timescale separation assumptions or singular perturbation, and a nonlinear integral control term is designed to guarantee robustness against unstructured uncertainties in both rotational and translational dynamics.

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 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, and (iv) an integral control term is proposed to guarantee asymptotical convergence of tracking error variables in the presence of uncertainties, (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 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 Section II. Control systems are constructed at Sections III and IV, which are followed by numerical examples in Section V. Finally, experimental results are presented in Section VI.

## 2 Problem Formulation

Consider a rigid body with the mass and the moment of inertia , being transported with arbitrary number of quadrotors as shown in Figure 2. The location of the mass center of the rigid body is denoted by , and its attitude is given by , where the special orthogonal group is given by . We choose an inertial frame and body fixed frame attached to the payload. We also consider a body fixed frame attached to the -th quadrotor . In the inertial frame, the third axes points downward along the gravity and the other axes are chosen to form an orthonormal frame.

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

 ∥Δx∥∞≤δ, (0)

for a positive constant . Throughout this paper, the two norm of a matrix is denoted by . The standard dot product is denoted by for any .

### 2.1 Lagrangian

 ˙qij=ωij×qij=^ωijqij, (0) ˙R0=R0^Ω0, (0) ˙Ri=Ri^Ωi, (0)

where is the angular velocity of the -th link in the -th cable satisfying . Also, is the angular velocity of the payload and is the angular velocity of the -th quadrotor, expressed with respect to the corresponding body fixed frame. The hat map is defined by the condition that for all . More explicitly, for a vector , the matrix is given by

 ^a=⎡⎢⎣0−a3a2a30−a1−a2a10⎤⎥⎦. (0)

This identifies the Lie algebra with using the vector cross product in . The inverse of the hat map is denoted by the vee map, . The position of the -th quadrotor is given by

 xi=x0+R0ρi−ni∑a=1liaqia, (0)

where is the vector from the center of mass of the rigid body to the point that -th cable is connected to the rigid body. Similarly the position of the -th link in the cable connecting the -th quadrotor to the rigid body is given by

 xij=x0+R0ρi−ni∑a=j+1liaqia. (0)

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

 T= 12m0∥˙x0∥2+n∑i=1ni∑j=112mij∥˙xij∥2+12n∑i=1mi∥˙xi∥2 +12n∑i=1Ωi⋅JiΩi+12Ω0⋅J0Ω0. (0)

The gravitational potential energy is given by

 V=−m0ge3⋅x0−n∑i=1mige3⋅xi−n∑i=1ni∑j=1mijge3⋅xij, (0)

where it is assumed that the unit-vector points downward along the gravitational acceleration as shown at Figure 2. The corresponding Lagrangian of the system is .

### 2.2 Euler-Lagrange equations

Coordinate-free form of Lagrangian mechanics on the two-sphere and the special orthogonal group for various multi-body systems has been studied in [?, ?]. The key idea is representing the infinitesimal variation of in terms of the exponential map

 δRi=ddϵ∣∣∣ϵ=0Riexp(ϵ^ηi)=Ri^ηi, (0)

for . The corresponding variation of the angular velocity is given by . Similarly, the infinitesimal variation of is given by

 δqij=ξij×qij, (0)

for satisfying . Using these, we obtain the following Euler-Lagrange equations.

{prop}

The equations of motion for the proposed payload transportation system are as follows

 MT¨x0−n∑i=1ni∑j=1M0ijlij¨qij−n∑i=1MiTR0^ρi˙Ω0 =MTge3+n∑i=1(−fiRie3+Δxi)−n∑i=1MiTR0^Ω20ρi, (0) ¯J0˙Ω0+n∑i=1MiT^ρiRT0¨x0−n∑i=1ni∑j=1M0ijlij^ρiRT0¨qij =n∑i=1^ρiRT0(−fiRie3+MiTge3+Δxi)−^Ω0¯J0Ω0, (0) ni∑k=1M0ijlik^q2ij¨qik−M0ij^q2ij¨x0+M0ij^q2ijR0^ρi˙Ω0 =M0ij^q2ijR0^Ω20ρi−^q2ij(M0ijge3−fiRie3+Δxi), (0) JiΩi+Ωi×JiΩi=Mi+ΔRi. (0)

Here the total mass of the system and the mass of the -th quadrotor and its flexible cable are defined as

 MT=m0+n∑i=1MiT,MiT=ni∑j=1mij+mi, (0)

and the constants related to the mass of links are given as

 M0ij =mi+j−1∑a=1mia, (0)

The equations of motion can be rearranged in a matrix form as follow

 N˙X=P (0)

where the state vector with is given by

 X=[˙x0,Ω0,˙q1j,˙q2j,⋯,˙qnj]T, (0)

and matrix is defined as

 (0)

where the sub-matrices are defined as

 Nx0Ω0=−n∑i=1MiTR0^ρi;NΩ0x0=MTx0Ω0, Nx0i=−[M0i1li1I3,M0i2li2I3,⋯,M0iniliniI3], NΩ0i=−[M0i1li1^ρiRT0,M0i2li2^ρiRT0,⋯,M0inilini^ρiRT0], Nix0=−[M0i1^q2i1,M0i2^q2i2,⋯,M0ini^q2ini]T, NiΩ0=[M0i1^q2i1R0^ρi,M0i2^q2i2R0^ρi,⋯,M0ini^q2iniR0^ρi]T, (0)

and the sub-matrix is given by

 Nqqi=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣−M011li1I3M012li2^q2i2⋯M01nilini^q2iniM021li1^q2i1−M022li2I3⋯M02nilini^q2ini⋮⋮⋮M0ni1li1^q2i1M0ni2li2^q2i2⋯−M0niniliniI3⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦. (0)

The matrix is

 P=[Px0,PΩ0,P1j,P2j,⋯,Pnj]T, (0)

and sub-matrices of matrix are also defined as

 Px0 =MTge3+n∑i=1(−fiRie3+Δxi)−n∑i=1MiTR0^Ω20ρi, PΩ0 =−^Ω0¯J0Ω0+n∑i=1^ρiRT0(MiTge3−fiRie3+Δxi), Pij= −^q2ij(−fiRie3+M0ijge3+Δxi)+M0ij^q2ijR0^Ω20ρi +M0ij∥˙qij∥2qij.
• See Appendix missingA.

These equations are derived directly on a nonlinear manifold without any simplification. The dynamics of the payload, flexible cables, and quadrotors are considered explicitly, and they avoid singularities and complexities associated to local coordinates.

## 3 Control System Design for Simplified Dynamic Model

### 3.1 Control Problem Formulation

Let be the desired position of the payload. The desired attitude of the payload is considered as , and the desired direction of links is aligned along the vertical direction. The corresponding location of the -th quadrotor at this desired configuration is given by

 xid=x0d+ρi−ni∑a=1liae3. (0)

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

### 3.2 Simplified Dynamic Model

Control forces for each quadrotor is given by for the given equations of motion (2.2), (2.2), (2.2), (2.2). As such, the quadrotor dynamics is under-actuated. 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 . 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 term by a fictitious control input , and design an expression for to asymptotically stabilize the desired equilibrium. In other words, we assume that the attitude of the quadrotor can be instantaneously changed. Also are ignored in the simplified dynamic model. The effects of the attitude dynamics are incorporated at the next section.

### 3.3 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 and , respectively. Also, we have and . In this equilibrium configuration, the control input for the -th quadrotor is

 uid=−fidRide3, (0)

where the total thrust is .

The variation of is given by

 δx0=x0−x0d, (0)

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

 δR0=R0d^η0=^η0,

for . The variation of can be written as

 δqij=ξij×e3, (0)

where with . The variation of is given by with . Therefore, the third element of each of and 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 , where . The variation of the control input , is given as .

{prop}

The linearized equations of the simplified dynamic model are given by

 M¨x+Gx=Bδu+g(x,˙x), (0)

where corresponds to the higher order term and the state vector with is given by

 x=[δx0,η0,CTξ1j,CTξ2j,⋯,CTξnj],

and . The matrix are defined as

where the sub-matrices are defined as

 Mx0Ω0=−n∑i=1MiT^ρi;MΩ0x0=MTx0Ω0, Mx0i=[M0i1li1^e3C,M0i2li2^e3C,⋯,M0inilini^e3C], MΩ0i=[M0i1li1^ρiC,M0i2li2^ρiC,⋯,M0inilini^ρiC], Mix0=−[M0i1CT^e3,M0i2CT^e3,⋯,M0iniCT^e3], (0) MiΩ0=[M0i1CT^e3^ρi,M0i2CT^e3^ρi,⋯,M0iniCT^e3^ρi], (0)

and the sub-matrix is given by

 Mqqi=⎡⎢ ⎢ ⎢ ⎢ ⎢⎣Mi11li1I2Mi12li2I2⋯Mi1niliniI2Mi21li1I2Mi22li2I2⋯Mi2niliniI2⋮⋮⋮Mini1li1I2Mini2li2I2⋯MininiliniI2⎤⎥ ⎥ ⎥ ⎥ ⎥⎦. (0)

The matrix is defined as

 G=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣0000000GΩ0Ω0000000G1000000G200⋮⋮⋮⋮⋮⋮00000Gn⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦,

where and the sub-matrices are

 Gi=diag[(−MiT−m0n+M0ij)ge3I2].

The matrix is given by

 B=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣I3I3⋯I3^ρ1^ρ2⋯^ρnBB0000BB00⋮⋮⋮⋮000BB⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦,

where .

• See Appendix missingB.

We present the following PD-type control system for the linearized dynamics

 δui= −Kxix−K˙xi˙x, (0)

for controller gains . Provided that (3.3) is controllable, we can choose the combined controller gains , such that the equilibrium is asymptotically stable for the linearized equations [?]. Then, the equilibrium becomes asymptotically stable for the nonlinear Euler-Lagrange equation. The controlled linearized system can be written as

 ˙z1=\mathdsAz1+\mathdsB(BX+g(x,˙x)), (0)

where and

 \mathdsA=[0I−M−1(G+BKx)−M−1BK˙x],\mathdsB=[0M−1]. (0)

We can also choose and such that is Hurwitz. Then for any positive definite matrix , there exist a positive definite and symmetric matrix such that according to [?, Thm 3.6].

## 4 Control System Design for the Full Dynamic Model

The control system designed at the previous section is based on a simplifying assumption that each quadrotor can generates 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 the previous section. Also in the full dynamics model, we considers the in the control design and introduce a new integral term to eliminate the disturbances and uncertainties. The central idea is that the attitude of the quadrotor is controlled such that its total thrust direction , corresponding to the third body-fixed axis, asymptotically follows the direction of the fictitious control input . By choosing the total thrust magnitude properly, we can guarantee asymptotical stability for the full dynamic model.

Let be the ideal total thrust of the -th quadrotor that asymptotically stabilize the desired equilibrium. Therefor, we have

 Ai=uid+δui=−Kxix−K˙xi˙x−Kzsatσ(ex)+uid, (0)

where and are the total thrust and control input of each quadrotor at its equilibrium respectively. where the following integral term is added to eliminate the effect of disturbance in the full dynamic model

 ex=∫t0(P\mathdsB)Tz1(τ)dτ, (0)

where is an integral gain. For a positive constant , a saturation function is introduced as

 satσ(y)=⎧⎨⎩σif y>σyif −σ≤y≤σ−σif y<−σ.

If the input is a vector , then the above saturation function is applied element by element to define a saturation function for a vector. From the desired direction of the third body-fixed axis of the -th quadrotor, namely , is given by

 b3i=−Ai∥Ai∥. (0)

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 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 exactly. Instead, its projection onto the plane normal to is followed, and the desired direction of the second body-fixed axis is chosen to constitute an orthonormal frame [?]. More explicitly, the desired attitude of the -th quadrotor is given by

 Ric=[−(^b3i)2b1i∥(^b3i)2b1i∥^b3ib1i∥^b3ib1i∥b3i], (0)

which is guaranteed to be an element of . The desired angular velocity is obtained from the attitude kinematics equation, .

Define the tracking error vectors for the attitude and the angular velocity of the -th quadrotor as

 eRi=12(RTicRi−RTiRic)∨,eΩi=Ωi−RTiRicΩic, (0)

and a configuration error function on as follows

 Ψi=12tr[I−RTicRi]. (0)

The thrust magnitude is chosen as the length of , projected on to , and the control moment is chosen as a tracking controller on :

 fi= −Ai⋅Rie3, (0) Mi= −kReRi−kΩeΩi−kIeIi +(RTiRciΩci)∧JiRTiRciΩci+JiRTiRci˙Ωci, (0)

where , and are positive constants and the following integral term is introduced to eliminate the effect of fixed disturbance

 eIi=∫t0eΩi(τ)+c2eRi(τ)dτ, (0)

where