Geometric Stabilization of a Quadrotor UAV with a PayloadConnected by Flexible Cable

# Geometric Stabilization of a Quadrotor UAV with a Payload Connected by Flexible Cable

## Abstract

This paper deals with dynamics and control of a quadrotor UAV with a payload that is connected via a flexible cable, which is modeled as a system of serially-connected links. It is shown that a coordinate-free form of equations of motion can be derived for an arbitrary number of links according to Lagrangian mechanics on a manifold. A geometric nonlinear control system is also presented to asymptotically stabilize the position of the quadrotor while aligning the links to the vertical direction. These results will be particularly useful for aggressive load transportation that involves deformation of the cable. The desirable properties are illustrated by a numerical example and a preliminary experimental result.

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## 1 Introduction

Quadrotor unmanned aerial vehicles (UAV) have been envisaged for various applications such as surveillance or mobile sensor networks as well as for educational purposes. Areal transportation of a cable-suspended load has been studied traditionally for helicopters [7, 8]. Recently, small-size single or multiple autonomous vehicles are considered for load transportation and deployment [9, 10, 11, 12], and trajectories with minimum swing of payload are generated [13, 14, 15].

However, these results are based on the common assumption that the cable connecting a quadrotor UAV to a payload is always taut. Therefore, they cannot be applied to aggressive, rapid load transportations where the cable is deformed or the tension along the cable is low, thereby restricting its applicability. It is challenging to incorporate the effects of a deformable cable, since the dimension of the configuration space becomes infinite. Finite element approximation of a cable often yields complicated equations of motion that make dynamic analysis and controller design extremely difficult.

Recently, a coordinate-free form of the equations of motion for a chain pendulum connected a cart that moves on a horizontal plane is presented according to Lagrangian mechanics on a manifold [19]. By following the similar approach, in this paper, the cable is modeled as an arbitrary number of links with different sizes and masses that are serially-connected by spherical joints, as illustrated at Figure 1. The resulting configuration manifold is the product of the special Euclidean group for the position and the attitude of the quadrotor, and a number of two-spheres that describe the direction of each link. We present Euler-Lagrange equations of the presented quadrotor model that are globally defined on the nonlinear configuration manifold.

The second part of this paper deals with nonlinear control system development. Quadrotor UAV is underactuated as the direction of the total thrust is always fixed relative to its body. By utilizing geometric control systems for quadrotor [4, 20, 21], we show that the hanging equilibrium of the links can be asymptotically stabilized while translating the quadrotor to a desired position. In contrast to existing papers where the force and the moment exerted by the payload to the quadrotor are considered as disturbances, the control systems proposed in this paper explicitly consider the coupling effects between the cable/load dynamics and the quadrotor dynamics.

Another distinct feature is that the equations of motion and the control systems are developed directly on the nonlinear configuration manifold in a coordinate-free fashion. This yields remarkably compact expressions for the dynamic model and controllers, compared with local coordinates that often require symbolic computational tools due to complexity of multibody systems. Furthermore, singularities of local parameterization are completely avoided to generate agile maneuvers in a uniform way.

This paper is organized as follows. A dynamic model and control systems are presented at Sections 2 through 4. The desirable properties of the proposed control system are illustrated by a numerical example at Section 5 and preliminary experimental results are given at Section 6.

## 2 Dynamic Model of a Quadrotor with a Flexible Cable

Consider a quadrotor UAV with a payload that is connected via a chain of links, as illustrated at Figure 1. The inertial frame is defined by the unit vectors , , and , and the third axis corresponds to the direction of gravity. Define a body-fixed frame whose origin is located at the center of mass of the quadrotor, and its third axis is aligned to the axis of symmetry.

The location of the mass center, and the attitude of the quadrotor are denoted by and , respectively, where the special orthogonal group is . A rotation matrix represents the linear transformation of a representation of a vector from the body-fixed frame to the inertial frame.

The dynamic model of the quadrotor is identical to [4]. The mass and the inertia matrix of the quadrotor are denoted by and , respectively. The quadrotor can generate a thrust with respect to the inertial frame, where is the total thrust magnitude. It also generates a moment with respect to its body-fixed frame. The pair is considered as control input of the quadrotor.

Let be the unit-vector representing the direction of the -th link, measured from the quadrotor toward the payload, where the two-sphere is the manifold of unit-vectors in , i.e., . For simplicity, we assume that the mass of each link is concentrated at the outboard end of the link, and the point where the first link is attached to the quadrotor corresponds to the mass center of the quadrotor. The mass and the length of the -th link are defined by and , respectively. Thus, the mass of the payload corresponds to . The corresponding configuration manifold of this system is .

Next, we define the kinematics equations. Let be the angular velocity of the quadrotor represented with respect to the body-fixed frame, and let be the angular velocity of the -th link represented with respect to the inertial frame. The angular velocity is normal to the direction of the link, i.e., . The kinematics equations are given by

 ˙R =R^Ω, (1) ˙qi =ωi×qi=^ωiqi, (2)

where the hat map is defined by the condition that for any , and it transforms a vector in to a skew-symmetric matrix. More explicitly, it is given by

 ^x=⎡⎢⎣0−x3x2x30−x1−x2x10⎤⎥⎦ (3)

for . The inverse of the hat map is denoted by the vee map .

Throughout this paper, the 2-norm of a matrix is denoted by , and the dot product is denoted by .

### 2.1 Lagrangian

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

 TQ=12m∥˙x∥2+12Ω⋅JΩ. (4)

Let be the location of in the inertial frame. It can be written as

 xi=x+i∑a=1laqa. (5)

Then, the kinetic energy of the links is given by

 TL =12n∑i=1mi∥˙x+i∑a=1la˙qa∥2 =12n∑i=1mi∥˙x∥+˙x⋅n∑i=1n∑a=imali˙qi+12n∑i=1mi∥i∑a=1la˙qa∥2. (6)

From (4) and (6), the total kinetic energy can be written as

 T =12M00∥˙x∥2+˙x⋅n∑i=1M0i˙qi+12n∑i,j=1Mij˙qi⋅˙qj +12ΩTJΩ, (7)

where the inertia values are given by

 M00=m+n∑i=1mi,M0i=n∑a=imali,Mi0=M0i, Mij=⎧⎨⎩n∑a=max{i,j}ma⎫⎬⎭lilj, (8)

for . The gravitational potential energy is given by

 V =−mgx⋅e3−n∑i=1migxi⋅e3 =−n∑i=1n∑a=imaglie3⋅qi−M00ge3⋅x, (9)

From (7) and (9), the Lagrangian is .

### 2.2 Euler-Lagrange equations

Coordinate-free form of Lagrangian mechanics on the two-sphere and the special orthogonal group for various multibody systems has been studied in [22, 23]. The key idea is representing the infinitesimal variation of in terms of the exponential map:

 δR=ddϵ∣∣∣ϵ=0expR(ϵ^η)=R^η, (10)

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

 δqi=ξi×qi, (11)

for satisfying . This lies in the tangent space as it is perpendicular to . Using these, the Euler-Lagrange equations can be derived as follows.

###### Proposition 1

Consider a quadrotor with a cable-suspended payload whose Lagrangian is given by (7) and (9). The Euler-Lagrange equations on are as follows:

 M00¨x+n∑i=1M0i¨qi=−fRe3+M00ge3, (12) Mii¨qi−^q2i(Mi0¨x+n∑j=1j≠iMij¨qj) =−Mii∥˙qi∥2qi−n∑a=imagli^q2ie3, (13) J˙Ω+^ΩJΩ=M, (14)

where is defined at (8). Equations (12) and (13) can be rewritten in a matrix form as follows:

 ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣M00M01M02⋯M0n−^q21M10M11I3−M12^q21⋯−M1n^q21−^q22M20−M21^q22M22I3⋯−M2n^q22⋮⋮⋮⋮−^q2nMn0−Mn1^q2n−Mn2^q2n⋯MnnI3⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣¨x¨q1¨q2⋮¨qn⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦ =⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣−fRe3+M00ge3−∥˙q1∥2M11q1−∑na=1magl1^q21e3−∥˙q2∥2M22q2−∑na=2magl2^q22e3⋮−∥˙qn∥2Mnnqn−mngln^q2ne3⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦. (15)

Or equivalently, it can be written in terms of the angular velocities as

 ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣M00−M01^q1−M02^q2⋯−M0n^qn^q1M10M11I3−M12^q1^q2⋯−M1n^q1^qn^q2M20−M21^q2^q1M22I3⋯−M2n^q2^qn⋮⋮⋮⋮^qnMn0−Mn1^qn^q1−Mn2^qn^q2⋯MnnI3⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣¨x˙ω1˙ω2⋮˙ωn⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦ =⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣∑nj=1M0j∥ωj∥2qj−fRe3+M00ge3∑nj=2M1j∥ωj∥2^q1qj+∑na=1magl1^q1e3∑nj=1,j≠2M2j∥ωj∥2^q2qj+∑na=2magl2^q2e3⋮∑n−1j=1Mnj∥ωj∥2^qnqj+mngln^qne3⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦, (16) ˙qi=ωi×qi. (17)
{proof}

See Appendix A. These provide a coordinate-free form of the equations of motion for the presented quadrotor UAV that is uniformly defined for any number of links , and that is globally defined on the nonlinear configuration manifold. Compared with equations of motion derived in terms of local coordinates, such as Euler-angles, these provide a compact form of equations that are suitable for control system design.

## 3 Control System Design for a Simplified Dynamic Model

### 3.1 Control Problem Formulation

Let be a fixed desired location of the quadrotor UAV. Assuming that all of the links are pointing downward, i.e., , the resulting location of the payload is given by . We wish to design the control force and the control moment such that this hanging equilibrium configuration at the desired location becomes asymptotically stable.

### 3.2 Simplified Dynamic Model

For the given equations of motion (12) for , the control force is given by . This implies that the total thrust magnitude can be arbitrarily chosen, but the direction of the thrust vector is always along the third body-fixed axis. Also, the rotational attitude dynamics of the quadrotor is not affected by the translational dynamics of the quadrotor or the dynamics of links.

In this section, we replace the term of (12) by a fictitious control input , and design an expression for to asymptotically stabilize the desired equilibrium. This is equivalent to assuming that the attitude of the quadrotor can be instantaneously changed. The effects of the attitude dynamics are studied at the next section.

In short, the equations of motion for the simplified dynamic model considered in the section are given by

 M00¨x+n∑i=1M0i¨qi=u+M00ge3, (18)

and (13).

### 3.3 Linear Control System

The fictitious control input is designed from the linearized dynamics about the desired hanging equilibrium. The variation of and are given by

 δx=x−xd,δu=u−M00ge3. (19)

From (11), the variation of from the equilibrium can be written as

 δqi=ξi×e3, (20)

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 equation, i.e., the state vector of the linearized equation is composed of , where .

###### Proposition 2

The linearized equations of the simplified dynamic model (18) and (13) can be written as follows:

 M¨x+Gx=Bδu, (21)

or equivalently

 [MxxMxqMqxMqq][δ¨x¨xq]+[0303×2n02n×3Gqq][δxxq]=[I302n×3]δu,

where the corresponding sub-matrices are defined as

 xq =[CTξ1;…;CTξn], Mxx =M00I3, Mxq =[−M01^e3C−M02^e3C⋯−M0n^e3C], Mqx =MTxq, Mqq =⎡⎢ ⎢ ⎢ ⎢⎣M11I2M12I2⋯M1nI2M21I2M22I2⋯M2nI2⋮⋮⋮Mn1I2Mn2I2⋯MnnI2⎤⎥ ⎥ ⎥ ⎥⎦, Gqq =diag[n∑a=1magl1I2,…,mnglnI2].
{proof}

See Appendix A.

For the linearized dynamics (21), the following control system is chosen:

 δu =−kxδx−k˙xδ˙x−n∑a=1kqiCT(e3×qi)−kωiCTδωi =−Kxx−K˙x˙x, (22)

for controller gains and . Provided that (21) is controllable, we can choose the controller gains such that the equilibrium is asymptotically stable for the linearized equation (21). Then, the equilibrium becomes asymptotically stable for the nonlinear Euler-Lagrange equation (18) and (13[24].

## 4 Controller Design for a Quadrotor with a Flexible Cable

The control system designed in the previous section is generalized to the full dynamic model that includes the attitude dynamics. The central idea is that the attitude of the quadrotor is controlled such that its total thrust direction , that corresponds 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.

### 4.1 Controller Design

Let be the ideal total thrust of the quadrotor system that asymptotically stabilize the desired equilibrium. From (22), we have

 Extra open brace or missing close brace (23)

The desired direction of the third body-fixed axis is given by

 b3c=−A∥A∥. (24)

This provides a two-dimensional constraint for the desired attitude of quadrotor, and there is additional one-dimensional degree of freedom that corresponds to rotation about the third body-fixed axis, i.e., yaw angle. A desired direction of the first body-fixed axis, namely is introduced to resolve it, and it is projected onto the plane normal to . The desired direction of the second body-fixed axis is chosen to constitute an orthonormal frame. More explicitly, the desired attitude is given by

 Rc=⎡⎢⎣−^b23cb1d∥^b23cb1d∥,^b3cb1d∥^b3cb1d∥,b3c⎤⎥⎦, (25)

which is guaranteed to lie in by construction, assuming that is not parallel to . The desired angular velocity is obtained by the attitude kinematics equation:

 Ωc=(RTc˙Rc)∨. (26)

Next, we introduce the tracking error variables for the attitude and the angular velocity as follows:

 eR =12(RTcR−RTRc)∨, (27) eΩ =Ω−RTRcΩc. (28)

The thrust magnitude and the moment vector of quadrotor are chosen as

 f =−A⋅Re3, (29) M =−kRϵ2eR−kΩϵeΩ+Ω×JΩ −J(^ΩRTRcΩc−RTRc˙Ωc), (30)

where are positive constants.

###### Proposition 3

Consider the full dynamics model defined by (12), (13), (14), (1), and (2). Control inputs are designed as (29) and (30), where the desired control force is given by (23). Then, there exist , such that for all , the desired equilibrium is exponentially stable.

{proof}

See Appendix A.

## 5 Numerical Example

The desirable properties of the proposed control system are illustrated by a numerical example. Properties of a quadrotor are chosen as

 m=0.5kg,J=diag[0.557,0.557,1.05]×10−2kgm2.

Five identical links with , , and are considered. Controller parameters are selected as follows: , , , , and .

The desired location of the quadrotor is selected as . The initial conditions for the quadrotor are given by

 x(0)=[0.6;−0.7;0.2], ˙x(0)=03×1, R(0)=I3×3,Ω(0)=03×1.

The initial direction of the links are chosen such that the cable is curved along the horizontal direction, as illustrated at Figure a, and the initial angular velocity of each link is chosen as zero.

We define the following two error functions to the performance of the proposed control system:

 eq=n∑i=1∥qi−e3∥,eω=n∑i=1∥ωi∥ (31)

Simulation results are illustrated at Figure 2, where the position of the quadrotor converges to the desired value , while reducing the direction error and the angular velocity error of the link. The corresponding maneuvers of the quadrotor and the links are illustrated by snapshots at Figure 3.

## 6 Preliminary Experimental Results

Preliminary experimental results are presented. A quadrotor UAV is developed with the following configuration, as illustrated at Figure 4:

• Gumstix Overo computer-in-module (OMAP 600MHz processor), running a non-realtime Linux operating system. It is connected to a ground station via WIFI.

• Microstrain 3DM-GX3 attitude sensor, connected to Gumstix via UART.

• BL-CTRL 2.0 motor speed controller, connected to Gumstix via I2C.

• Roxxy 2827-35 Brushless DC motors.

• XBee RF module, connected to Gumstix via UART.

The mass of the quadrotor is . A payload with mass of is attached to the quadrotor via a cable of length . For the given preliminary experiments, the cable is modeled by a single link, i.e., . The locations of the quadrotor and the payload are measured by a VICON motion capture system, and they are transferred to the onboard computer module via XBee to estimate the full state and compute the control inputs.

Two cases are considered and compared. For the first case, a position control system developed in [21], for quadrotor UAV that does not include the dynamics of the payload and the link, is applied to hover the quadrotor at the desired location, and the second case, the proposed control system is used.

Experimental results are shown at Figures 5 and 6. The position of the quadrotor and the payload is compared with the desired position of the quadrotor at Figure 5, and the deflection angle of the link from the vertical direction are illustrated at Figure 6. It is shown that the proposed control system reduces the undesired oscillation of the link effectively, compared with the quadrotor position control system that does not include the dynamics of links and payload.1

## 7 Conclusions

Euler-Lagrange equations have been used for the quadrotor and the chain pendulum to model a flexible cable transporting a load in 3D space. These derivations developed in a remarkably compact form which allow us to choose arbitrary number and any configuration of the links. We developed a geometric nonlinear controller to stabilize the links below the quadrotor in the equilibrium position from an any chosen initial condition. We expanded these derivations in such way that there is no need of using local angle coordinate and this advantageous technique signalize our derivations.

## Appendix

### .1 Proof for Proposition 1

From (7) and (9), the Lagrangian is given by

 L =12M00∥˙x∥2+˙x⋅n∑i=1M0i˙qi+12n∑i=1mi∥i∑a=1la˙qa∥2 +n∑i=1n∑a=imaglie3⋅qi+M00ge3⋅x+12ΩTJΩ, (32)

The derivatives of the Lagrangian are given by

 DxL =M00ge3, D˙xL =M00˙x+n∑i=1M0i˙qi,

where represents the derivative of with respect to . From the variation of the angular velocity given after (10), we have

 DΩL⋅δΩ=JΩ⋅(˙η+Ω×η)=JΩ⋅˙η−η⋅(Ω×JΩ). (33)

Similarly from (11), the derivative of the Lagrangian with respect to is given by

 DqiL⋅δqi=n∑a=imaglie3⋅(ξi×qi)=−n∑a=imagli^e3qi⋅ξi.

The variation of is given by

 δ˙qi=˙ξi×qi+ξi×qi.

Using this, the derivative of the Lagrangian with respect to is given by

 D˙qiL⋅δ˙qi=(Mi0˙x+n∑j=1Mij˙qj)⋅δ˙qi =(Mi0˙x+n∑j=1Mij˙qj)⋅(˙ξi×q+ξi×˙qi) =^qi(Mi0˙x+n∑j=1Mij˙qj)⋅˙ξi+^˙qi(Mi0˙x+n∑j=1Mij˙qj)⋅ξi.

Let be the action integral, i.e., . From the above expressions for the derivatives of the Lagrangian, the variation of the action integral can be written as

 δG =∫tft0{M00˙x+n∑i=1M0i˙qi}⋅δ˙x+M00ge3⋅δx +n∑i=1{^qi(Mi0˙x+n∑j=1Mij˙qj)}⋅˙ξi +n∑i=1{^˙qi(Mi0˙x+n∑j=1Mij˙qj)−n∑a=imagli^e3qi}⋅ξi +JΩ⋅˙η−η⋅(Ω×JΩ)dt.

Integrating by parts and using the fact that variations at the end points vanish, this reduces to

 δG =∫tft0{M00ge3−M00¨x−n∑i=1M0i¨qi}⋅δx +n∑i=1{−^qi(Mi0¨x+n∑j=1Mij¨qj)−n∑a=imagli^e3qi}⋅ξi −η⋅(J˙Ω+Ω×JΩ)dt.

According to the Lagrange-d’Alembert principle, the variation of the action integral is equal to the negative of the virtual work done by the external force and moment, namely

 −∫tft0−fRe3⋅δx−M⋅ηdt.

Therefore, we obtain (12) and (14). As is perpendicular to , we also have

 −^q2i(Mi0¨x+n∑j=1Mij¨qj)+n∑a=imagli^q2ie3=0. (34)

Equation (34) is rewritten to obtain an explicit expression for . As , we have . Using this, we have

 −^q2i¨qi=−(qi⋅¨qi)qi+(qi⋅qi)¨qi=(˙qi⋅˙qi)qi+¨qi.

Substituting this into (34), we obtain (13).

This can be slightly rewritten in terms of the angular velocities. Since for the angular velocity satisfying , we have

 ¨qi =˙ωi×qi+ωi×(ωi×qi) =˙ωi×qi−∥ωi∥2qi=−^qi˙ωi−∥ωi∥2qi.

Using this and the fact that , we obtain (16).

The variations of and are given by (19) and (20). From the kinematics equation , is given by

 δ˙qi=˙ξi×e3=δωi×e3+0×(ξi×e3)=δωi×e3.

Since both sides of the above equation is perpendicular to , this is equivalent to , which yields

 ˙ξ−(e3⋅˙ξ)e3=δωi−(e3⋅δωi)e3.

Since , we have . As from the constraint, we obtain the linearized equation for the kinematics equation:

 ˙ξi=δωi. (35)

Substituting these into (16), and ignoring the higher order terms, we obtain (21).

### .2 Proof for Proposition 3

This proof is based on singular perturbation [24] and the attitude tracking control system developed in [4]. Let . The error dynamics for can be written as The error dynamics for are given by

 ˙eRi =12(tr[RTiRci]I−RTiRci)eΩi, Ji˙eΩi =−kRϵ2eRi−kΩϵeΩi.

Define to rewrite these as the standard form of singular perturbation:

 ϵ˙¯eR =12(tr[RTRc]I−RTiRc)eΩ, ϵ˙eΩ =J−1(−kR¯eR−kΩeΩ).

The right-hand side of the above equations has an isolated root of , and they correspond to the boundary-layer system. And, the origin of the boundary-layer system is exponentially stable according to [4, Proposition 1].

More explicitly, define a configuration error function on as follows:

 ΨR=12tr[I−RTcR].

Consider a domain given by . Define a Lyapunov function,

 W =12eΩ⋅JeΩ+kRϵ2ΨR+c3ϵeR⋅eΩ,

where