research article rotor-flying manipulator: modeling, analysis,...

Research ArticleRotor-Flying Manipulator: Modeling, Analysis, and Control

Bin Yang,1,2 Yuqing He,1 Jianda Han,1 and Guangjun Liu1,3

1 State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China2University of Chinese Academy of Sciences, Beijing 100049, China3Department of Aerospace Engineering, Ryerson University, Toronto, ON, Canada M5B 2K3

Correspondence should be addressed to Yuqing He; [email protected]

Received 14 December 2013; Revised 24 February 2014; Accepted 26 February 2014; Published 4 May 2014

Academic Editor: Rongni Yang

Copyright © 2014 Bin Yang et al.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Equipping multijoint manipulators on a mobile robot is a typical redesign scheme to make the latter be able to actively influencethe surroundings and has been extensively used for many ground robots, underwater robots, and space robotic systems. However,the rotor-flying robot (RFR) is difficult to be made such redesign. This is mainly because the motion of the manipulator will bringheavy coupling between itself and the RFR system, which makes the system model highly complicated and the controller designdifficult.Thus, in this paper, the modeling, analysis, and control of the combined system, called rotor-flying multijoint manipulator(RF-MJM), are conducted. Firstly, the detailed dynamics model is constructed and analyzed. Subsequently, a full-state feedbacklinear quadratic regulator (LQR) controller is designed through obtaining linearized model near steady state. Finally, simulationsare conducted and the results are analyzed to show the basic control performance.

1. Introduction

Rotor-flying robot (RFR) has been researched for severaldecades and achieved great development. To date, RFRs haveshown their priority inmany applications, such as search, res-cue, and surveillance [1–6]. However, these applications areoften passive. That means the RFR system can’t manipulatethe interested objects by a direct physical interaction.

Most recently, this problem has attained more and moreattentions from many researches. For example, someresearchers suggest to equip a gripper on the RFR systemso that the RFR system can grasp as shown Figure 1 [7–9].But it still has some disadvantages including that (1) themanipulation can only be implemented through controllingthe attitude of the RFR system. However, the precise controlof the RFR system is difficult due to its complicated dynamics,and the precise manipulation is impossible. (2) With thisstructure, in order to manipulate an object, the RFR mustapproach it. This, however, may both bring the so-calledground effect and “blow” the object, which makes the precisecontrol much more difficult.

In this paper, a new system structure as shown in Figure 2is proposed. The system is composed of an RFR system and

a multijoint manipulator and thus called rotor-flying mul-tijoint manipulator (RF-MJM). Compared to the structurein Figure 1, the multiple-joint manipulator can be used toregulate position and attitude of the end-gripper. This is veryuseful to compensate the control imprecision of the RFRsystem and makes precise manipulation much easier.

However, it is obvious that the system shown in Figure 2is very difficult to be controlled. This is mainly because theRFR system itself is very sensitive to the external disturbance(force/moment), especially for the time varying disturbance,for example, from amovingmanipulator as the new proposedRF-MJM system.Thus, it is very important and necessary forus to study the detailed model that can describe the couplingbetween the RFR and the manipulator. What is more, toconstruct a full-state high fidelity dynamics model is also abenefit for optimizing the design parameters, for example,the mass, the joint number, and the configuration of themanipulator, and testing the designed control algorithms.

Thus in this paper, the dynamics model of the RF-MJM system is constructed and analyzed to show its basicperformance.Moreover, the linear LQR controller is designedto test the basic closed loop performance of it. The maincontributions of this paper are as the following three aspects:

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 492965, 13 pageshttp://dx.doi.org/10.1155/2014/492965

2 Mathematical Problems in Engineering

(a) (b)

Figure 1: RFR system designed by Yale University (a) and DLR (b).

Figure 2: Sketch of the RF-MJM system.

(1) a detailed full-state high fidelity nonlinear dynamicsmodel of the RF-MJM system is constructed through usingEuler-Lagrangemethod; (2) the dynamics couplings betweenthe RFR and the manipulator are analyzed in detail, whichis good for the optimal design of the system configuration;(3) LQR controller is designed based on the linearized systemmodel and simulations are conducted to show the basiccontrol performance of the new proposed system.

2. Dynamics Model of RF-MJM

The dynamics model of the RF-MJM is composed of threeparts as shown in Figure 3: the body dynamics model, themid-dynamics model, and the actuator model. The bodydynamics model describes the relation between the motionstate and the external force/torque exerting on the bodyof the robot; the mid-dynamics model represents how theforce/torque is produced, that is, the aerodynamics of theRFR, and the torque from the manipulator joint motor, whilethe actuator model depicts the dynamical characteristic ofthe actuator, for example, the motors of the manipulator andthe steering engine of the RFR. For the RF-MJM system,

Body dynamics model

Actuator model

Mid-dynamics model

Force/torque exerting on the robot

State of actuator

Motion state of robot

System output

System input

Figure 3: Model structure of the RF-MJM system.

the coupling between the RFR and the manipulator willmainly influence the body dynamics model, which, then, willbe discussed in this paper.

ARF-MJM system is actually amultilink system shown asin Figure 4,where the cube denotes theRFR and the ellipsoidsdenote the link of themanipulator; Σ

0, Σ𝐼, Σ𝐸, and Σ

𝑖are RFR

body-fixed reference frame, earth-fixed inertial frame, end-gripper frame, and the frame of the 𝑖th joint of manipulator;𝐽𝑖(𝑖 = 1, 2, . . . , 𝑛) denote the joint of the manipulator; 𝑝

𝑖

denotes its position vector in the frame of Σ𝐼; 𝐶

0and 𝐶

𝑖are

the position vector of the center of mass (COM) of the linkRFR and the link 𝑖; 𝑑

0and 𝑑

𝑖are the position vectors of 𝐶

0

and 𝐶𝑖; 𝑎𝑖is the vector from 𝐽

𝑖to 𝐶

𝑖; 𝑏0is the vector from

COM of the RFR to the first joint; 𝑏𝑖is vector from 𝐶

𝑖to 𝐽

𝑖+1;

𝑛 is the number of the manipulator’s link.

2.1. Kinematics Model. In this paper, we suppose that boththe RFR system and the manipulator are rigid. Thus, thefollowing geometric relations are satisfied:

𝑑𝑖= 𝑑

0+ 𝑏

0+

𝑖−1

∑𝑘=1

(𝑎𝑘+ 𝑏

𝑘) + 𝑎

𝑖. (1)

Differentiate it with respect to time and we have

V𝑖= 𝑑

𝑖= V

0+ 𝜔

0× (𝑑

𝑖− 𝑑

0) +

𝑖

∑𝑘=1

{

𝑘𝑘× (𝑑

𝑖− 𝑝

𝑘) ��𝑘} ,

(2)

Mathematical Problems in Engineering 3

Y0C0

Z0

X0 b0

b2

b k

b n

Ck

Cn

a k

a n

b1

d1

d2

a2

a1

C2

C1

dk

P1

P2

Pk

Pn

Pe

dn

d0

Σ0

ΣI

Σn

Σk

ΣE

Σ1

Σ2

J2

Jk

J1

Jn

YI

ZI

X10I

SystemCM

...

...

Figure 4: System structure of the RF-MJM system.

where V0and V

𝑖are the linear velocity of the COM of the RFR

and the link 𝑖, respectively; 𝜔0is the angular velocity of the

RFR in the frame of Σ𝐼; 𝜃𝑘is the angular position vector of

the 𝑘th joint; 𝑘𝑘denotes the unit vector of the axis 𝑍

𝑖of the

𝑖th joint frame; 𝑑𝑖, 𝑝

𝑘represent the position vectors of the

COM of the link 𝑖 and the 𝑘th joint. The angular velocity ofthe 𝑖th joint can be denoted as

𝜔𝑖= 𝜔

0+

𝑖

∑𝑘=1

𝑘𝑘𝜃𝑘. (3)

Combine (2) and (3), and the kinematic model of the RF-MJM system is

[V𝑖

𝜔𝑖

] = 𝐽𝑏𝑖[V0

𝜔0

] + 𝐽𝑚𝑖Θ, (4)

where 𝐽𝑏𝑖is the Jacobian matrix of the RFR system and has

the following form:

𝐽𝑏𝑖= [

𝐸 −𝑑0𝑖

0 𝐸] + [

𝐽𝑏V𝑖𝐽𝑏𝜔𝑖

] . (5)

In (5), 𝐸 is the unity matrix with proper dimension:

𝑑0𝑖= 𝑑

𝑖− 𝑑

0= [𝑑0𝑖,𝑥 𝑑

0𝑖,𝑦𝑑0𝑖,𝑧]

𝑇 (6)

and 𝑑0𝑖is the skew-symmetric matrix of the vector 𝑑

0𝑖; that

is,

𝑑0𝑖= [

[

0 −𝑑0𝑖,𝑧

𝑑0𝑖,𝑦

𝑑0𝑖,𝑧

0 −𝑑0𝑖,𝑥

−𝑑0𝑖,𝑦

𝑑0𝑖,𝑥

0

]

]

. (7)

𝐽𝑚𝑖in (4) is the Jacobianmatrix of themanipulator system

defined as

𝐽𝑚𝑖= [

𝑘1× (𝑑

𝑖− 𝑝

1) ⋅ ⋅ ⋅ 𝑘

𝑖× (𝑑

𝑖− 𝑝

𝑖)

𝑘1

⋅ ⋅ ⋅ 𝑘𝑖

] = [𝐽𝑚V𝑖𝐽𝑚𝜔𝑖

] . (8)

Also, we can transform the linear velocity of the RFR V0

into the velocity in the RFR body frame; that is,

V0= [

[

𝑐𝜃𝑐𝜑 𝑠𝜙𝑠𝜃𝑐𝜑 − 𝑐𝜙𝑠𝜑 𝑐𝜙𝑠𝜃𝑐𝜑 + 𝑠𝜙𝑠𝜑

𝑐𝜃𝑠𝜑 𝑐𝜙𝑐𝜑 + 𝑠𝜃𝑠𝜙𝑠𝜑 𝑐𝜙𝑠𝜃𝑠𝜑 − 𝑠𝜙𝑐𝜑

−𝑠𝜃 𝑠𝜙𝑐𝜃 𝑐𝜙𝑐𝜃

]

]

[

[

𝑢

V𝑤

]

]

, (9)


where 𝑐 and 𝑠 mean trigonometric function cos and sin,respectively;Φ = [𝜙, 𝜃, 𝜓]𝑇 represents the attitude of theRFR;[𝑢, V, 𝑤] is the linear velocity of the RFR in the RFR body fixedreference frame.

Similarly, the relation between the angular velocity in theframe of Σ

𝐼and that in the body frame is as follows:

𝜔0=

[[[[

[

1 sin𝜙 tan 𝜃 cos𝜙 tan 𝜃

0 cos𝜙 − sin𝜙

0 sin𝜙 sec 𝜃 cos𝜙 sec 𝜃

]]]]

]

[[[[

[

𝑝

𝑞

𝑟

]]]]

]

, (10)

where 𝑝, 𝑞, and 𝑟 are the components of the angular velocityalong the axes of the RFR body fixed reference frame.

2.2. Dynamics Model. In this section, the dynamics model ofthe RF-MJM system will be deduced using Euler-Lagrangemethod.

2.2.1. Kinetic Energy. Firstly, the kinetic energy of the systemcan be denoted as

𝐸𝑘=1

2

𝑛

∑𝑖=1

(𝜔𝑇

𝑖𝐼𝑖𝜔𝑖+ 𝑚

𝑖V𝑇𝑖V𝑖) , (11)

where𝑚𝑖and 𝐼

𝑖are the mass and the inertia tensor of the 𝑖th

part[10].Substitute (2) and (3) into (11), we have

𝐸𝑘=1

2

[[

[

V0

𝜔0

Θ

]]

]

𝑇

[[[

[

𝜛𝐸 𝜛𝑑𝑇

0𝑔𝐽𝑇𝜔

𝜛𝑑0𝑔 𝐻𝜔 𝐻𝜔𝜙

𝐽𝑇

𝑇𝜔𝐻𝑇

𝜔𝜙𝐻𝑚

]]]

]

[[

[

V0

𝜔0

Θ

]]

]

=1

2

[[

[

V0

𝜔0

Θ

]]

]

𝑇

𝐻[[

[

V0

𝜔0

Θ

]]

]

,

(12)

where 𝐻 is called the inertia matrix of the RF-MJM systemwith

𝐻𝑚=

𝑛

∑𝑖=1

(𝐽𝑇

𝑅𝑖𝐼𝐽𝑅𝑖+ 𝑚

𝑖𝐽𝑇

𝑇𝑖𝐽𝑇𝑖)

𝐽𝑇𝜔=

𝑛

∑𝑖=1

(𝑚𝑖𝐽𝑇𝑖)

𝐻𝜔=

𝑛

∑𝑖=1

(𝐼𝑖+ 𝑚

𝑖𝑑𝑇

0𝑖𝑑0𝑖) + 𝐼

0

𝐻𝜔𝜙=

𝑛

∑𝑖=1

(𝐼𝑖𝐽𝑅𝑖+ 𝑚

𝑖𝑑𝑇

0𝑖𝐽𝑇𝑖)

𝐽𝑅𝑖= [𝑘1 𝑘2 ⋅ ⋅ ⋅ 𝑘𝑖 0 ⋅ ⋅ ⋅ 0]

𝐽𝑇𝑖

= [𝑘1×(𝑑𝑖−𝑝1) 𝑘2× (𝑑𝑖−𝑝2) ⋅ ⋅ ⋅ 𝑘𝑖 × (𝑑𝑖− 𝑝𝑖) 0 ⋅ ⋅ ⋅ 0] ,

𝑑𝑔=∑𝑛

1=0𝑚𝑖𝑑𝑖

∑𝑛

1=0𝑚𝑖

.

(13)

2.2.2. Potential Energy. In this part, the potential energy dueto the gravity will be deduced. Firstly, based on Figure 2, (1)can be rewritten as

𝑑𝑖= 𝑑

0+

𝑖

∑𝑘=1

(𝐼𝐴𝑘−1𝐶𝑘−1,𝑘

−𝐼𝐴𝑘𝐶𝑘,𝑘) , (14)

where 𝐼𝐴0and 𝐼

𝐴𝑘denote the coordinate transformation

matrix from the frame Σ0and the frame Σ

𝑘to the frame

Σ𝐼, respectively; 𝐶

𝑘,𝑘is the position vector from joint 𝑖 to

the COM of the 𝑖th partdenoted in the 𝑖th joint coordinatesystem.

With (14), the potential energy due to gravity of the RF-MJM system can be easily obtained as follows:

𝐸𝑝= −

𝑛

∑𝑖=0

𝑚𝑖𝐺𝑇𝑑𝑖

=

𝑛

∑𝑖=0

𝑚𝑖𝐺𝑇(𝑑

0+

𝑖−1

∑𝑘=1

(𝐼𝐴𝑘−1𝐶𝑘−1,𝑘

−𝐼𝐴𝑘−1𝐶𝑘,𝑘)) ,

(15)

where

𝐺 = [

[

0

0

𝑔

]

]

(16)

and 𝑔 is the acceleration due to gravity.

2.2.3. Dynamics Model. The Euler-Lagrangian equation ofthe RF-MJM system is

𝐿 = 𝐸𝑘− 𝐸

𝑝. (17)

From (12), the kinetic energy can be rewritten as

𝐸𝑘=1

2[𝜛V𝑇

0𝐸V0+ 𝜛𝜔

𝑇

0𝑑0𝑔V0+ Θ

𝑇𝐽𝑇

𝑇𝜔V0

+ 𝜛V𝑇0𝑑𝑇

0𝑔𝜔0+ 𝜔

𝑇

0𝐻𝜔𝜔0+ Θ

𝑇𝐻𝑇

𝜔𝜙𝜔0

+ V𝑇0𝐽𝑇𝜔Θ + 𝜔

𝑇

0𝐻𝜔𝜙Θ + Θ

𝑇𝐻𝑚Θ] .

(18)


Thus we have

𝜕𝐸𝑘

𝜕 𝑞=

[[[[[[[

[

𝜕𝐸𝑘

𝜕V0

𝜕𝐸𝑘

𝜕𝜔0

𝜕𝐸𝑘

𝜕Θ

]]]]]]]

]

=[[[

[

𝜛𝐸V0+ 𝜛𝑑𝑇

0𝑔𝜔0+ 𝐽

𝑇𝜔Θ

𝐻𝜔𝜔0+ 𝜛𝑑

0𝑔V0+ 𝐻

𝜔𝜙Θ

𝐻𝑚Θ + 𝐻𝑇

𝜔𝜙𝜔0+ 𝐽𝑇

𝑇𝜔V0

]]]

]

, (19)

𝑑

𝑑𝑡(𝜕𝐸𝑘

𝜕 𝑞) =

[[[[[[[

[

𝑑


𝜕V0

)

𝑑


𝜕𝜔0

)

𝑑


𝜕Θ)

]]]]]]]

]

=

[[[[[[

[

𝜛𝐸V0+ 𝜛

𝑑𝑇

0𝑔𝜔0+ 𝜛𝑑𝑇

0𝑔��0+ 𝐽

𝑇𝜔Θ + 𝐽

𝑇𝜔Θ

��𝜔𝜔0+ 𝐻

𝜔��0+ 𝜛

𝑑0𝑔V0+ 𝜛𝑑

0𝑔V0+ ��

𝜔𝜙Θ + 𝐻

𝜔𝜙Θ

��𝑚Θ + 𝐻

𝑚Θ + ��𝑇

𝜔𝜙𝜔0+ 𝐻𝑇

𝜔𝜙��0+ 𝐽𝑇

𝑇𝜔V0+ 𝐽𝑇

𝑇𝜔V0

]]]]]]

]

, (20)

𝜕𝐸𝑘

𝜕𝑞=

[[[[[[[[

[

𝜕𝐸𝑎

𝜕𝑝+𝜕𝐸𝑏

𝜕𝑝+𝜕𝐸𝑐

𝜕𝑝

𝜕𝐸𝑎

𝜕Φ+𝜕𝐸𝑏

𝜕Φ+𝜕𝐸𝑐

𝜕Φ

𝜕𝐸𝑎

𝜕Θ+𝜕𝐸𝑏

𝜕Θ+𝜕𝐸𝑐

𝜕Θ

]]]]]]]]

]

, (21)

where

𝑞 =

[[[[

[

𝑋

Φ

Θ

]]]]

]

, 𝑞 =

[[[[

[

V0

𝜔0

Θ

]]]]

]

(22)

are the position vector and the velocity vector of the RF-MJMsystem, respectively;𝑋 is the position vector of the RFR; and

𝐸𝑎= 𝜛𝜔

𝑇

0𝑑0𝑔V0; 𝐸

𝑏=1

2𝜔𝑇

0𝐻𝜔𝜔0;

𝐸𝑐= (V𝑇

0𝐽𝑇𝜔+ 𝜔

𝑇

0𝐻𝜔𝜙+1

2Θ𝑇𝐻𝑚) Θ.

(23)

Similarly, with respect to the potential energy term, wehave

𝜕𝐸𝑝

𝜕𝑋=

𝑛

∑𝑖=0

𝑚𝑖

[[[[[[[[[

[

𝜕𝑔𝑇𝑑0

𝜕𝑥0

𝜕𝑔𝑇𝑑0

𝜕𝑦0

𝜕𝑔𝑇𝑑0

𝜕𝑧0

]]]]]]]]]

]

, (24)

𝜕𝐸𝑝

𝜕Θ= −(

𝑛

∑𝑖=0

𝑚𝑖𝜕𝑔𝑇

× (𝑑0+

𝑖

∑𝑗=1

(𝐼𝐴𝑗−1∗𝐶

𝑗−1,𝑗−𝐼𝐴𝑗∗𝐶

𝑗,𝑗)))

× (𝜕Θ)−1,

(25)

𝜕𝐸𝑝

𝜕Φ

= −

[[[[[[[

[

𝜕𝐸𝑝

𝜕𝜙𝜕𝐸𝑝

𝜕𝜃𝜕𝐸𝑝

𝜕𝜑

]]]]]]]

]

= −

[[[[[[[[[

[

−∑𝑛

𝑖=1𝑚𝑖𝜕𝑔𝑇∑

𝑖

𝑗=1(𝐼𝐴𝑗−1∗𝐶


𝑗,𝑗)

𝜕𝜙

−∑𝑛


𝑖



𝑗,𝑗)

𝜕𝜃−∑

𝑛


𝑖



𝑗,𝑗)

𝜕𝜑

]]]]]]]]]

]

.

(26)

Thus, the dynamics model of the RF-MJM system can beobtained through the following Euler-Lagrange equation:

𝑑

𝑑𝑡

𝜕𝐸𝑘

𝜕 𝑞−𝜕𝐸𝑘

𝜕𝑞+𝜕𝐸𝑃

𝜕𝑞= 𝜏. (27)

Substitute (20)–(21) and (24)–(26) into (27) and aftersimplifying we can obtain the following dynamics model ofthe RF-MJM system:

[𝐻𝑏𝐻𝑏𝑚

𝐻𝑇𝑏𝑚

𝐻𝑚

][��𝑏

Θ] + [

𝐶𝑏

𝐶𝑚

] + [𝐺𝑏

𝐺𝑚

] = [𝐹𝑝

𝜏𝑚

] , (28)

where

��𝑏= [

V0

��0

] . (29)


𝐶𝑏and𝐶

𝑚are theCoriolis and centrifugal force of the system;

𝐺𝑏and 𝐺

𝑚are the force due to gravity;

𝐹𝑝= [

𝐹𝐵

𝑀𝐵

] ; 𝐹𝐵=[[

[

𝐹𝑥

𝐹𝑦

𝐹𝑧

]]

]

; 𝑀𝐵=[[

[

𝐿

𝑀

𝑁

]]

]

(30)

are the force and moment produced by the RFR; the otherterms are defined as follows:

𝐻𝑏= [

𝐻𝑏11

𝐻𝑏12

𝐻𝑏21

𝐻𝑏22

]

𝐻𝑏11= 𝜛𝐸; 𝐻

𝑏12= 𝜛𝑑

𝑇

𝑜𝑔;

𝐻𝑏21= 𝜛𝑑

𝑜𝑔; 𝐻

𝑏22= 𝐻

𝜔

𝐻𝑏𝑚= [

𝐻𝑏𝑚V

𝐻𝑏𝑚𝜔

]

2×1

𝐻𝑏𝑚V= 𝐽

𝑇𝜔; 𝐻

𝑏𝑚𝜔= 𝐻

𝜔𝜙

𝐶𝑏= [

𝐶𝑏V

𝐶𝑏𝜔

]

𝐶𝑏V = 𝜛

𝑑𝑇

0𝑔𝜔0+ 𝐽

𝑇𝜔Θ − 𝜛𝜔

𝑇

0

𝜕𝑑0𝑔

𝜕𝑋V0− V𝑇

0

𝜕𝐽𝑇𝜔

𝜕𝑋Θ

− 𝜔𝑇

0

𝜕𝐻𝜔𝜙

𝜕𝑋Θ −

1

2(𝜔

𝑇

0

𝜕𝐻𝜔

𝜕𝑋𝜔0+ Θ

𝑇 𝜕𝐻𝑚

𝜕𝑋Θ)

𝐶𝑏𝜔= ��

𝜔𝜔0+ 𝜛

𝑑0𝑔V0+ ��

𝜔𝜙Θ − 𝜛𝜔

𝑇

0

𝜕𝑑0𝑔

𝜕ΦV0− V𝑇

0

𝜕𝐽𝑇𝜔

𝜕ΦΘ

− 𝜔𝑇

0

𝜕𝐻𝜔𝜙

𝜕ΦΘ −

1

2(𝜔

𝑇

0

𝜕𝐻𝜔

𝜕Φ𝜔0+ Θ

𝑇 𝜕𝐻𝑚

𝜕ΦΘ) .

𝐶𝑚=1

2𝐽𝑇

𝑇𝜔V0+1

2V𝑇0𝐽𝑇𝜔+1

2��𝑇

𝜔𝜙𝜔0+1

2𝜔𝑇

0��𝜔𝜙+ ��

𝑚Θ.

(31)

2.2.4. Extended Dynamics Model. When the manipulatorcontacts some external objects, the dynamics model (28)becomes

[𝐻𝑏𝐻𝑏𝑚

𝐻𝑇𝑏𝑚

𝐻𝑚

][

[

��𝑏

Θ

]

]

+ [

[

𝐶𝑏

𝐶𝑚

]

]

+ [

[

𝐺𝑏

𝐺𝑏

]

]

= [

[

𝐹𝑝

𝜏𝑚

]

]

+ [

[

𝐽𝑇𝑏

𝐽𝑇𝑚

]

]

𝐹𝑒,

(32)

where 𝐹𝑒is the force and torque exerting on the end of the

manipulator; 𝐽𝑏and 𝐽

𝑚are the Jacobian matrix defined as

𝐽𝑏= [

[

𝐸 −𝑝0𝑖

0 𝐸

]

]

; 𝑝0𝑖= 𝑝

𝑖− 𝑑

0;

𝐽𝑚= [

𝑘1× (𝑝

𝑒− 𝑝

1) 𝑘

2× (𝑝

𝑒− 𝑝

2) ⋅ ⋅ ⋅ 𝑘

𝑛× (𝑝

𝑒− 𝑝

𝑛)

𝑘1

𝑘2

⋅ ⋅ ⋅ 𝑘𝑛

] .

(33)

Furthermore, if we consider the aerodynamics of the RFRsystem, the force and moment produced by the RFR 𝐹

𝑝can

be denoted as the following mathematical equations [11]:

𝐹𝑋= −𝑇

𝑀sin 𝑎

1𝑠; 𝐹

𝑌= 𝑇

𝑀sin 𝑏

1𝑠− 𝑇

𝑇;

𝐹𝑍= −𝑇

𝑀cos 𝑎

1𝑠cos 𝑏

1𝑠;

𝐿 = −(𝜕𝐿𝑀

𝜕𝑏1𝑠

) 𝑏1𝑠− 𝑄

𝑀sin 𝑎

1𝑠;

𝑀 = (𝜕𝑀

𝑀

𝜕𝑎1𝑠

)𝑎1𝑠− 𝑄

𝑀sin 𝑏

1𝑠− 𝑄

𝑇;

𝑁 = −𝑄𝑀cos 𝑎

1𝑠cos 𝑏

1𝑠,

(34)

where 𝑇𝑀

and 𝑇𝑇are the forces derived from the main

rotor and tail rotor of the RFR, and 𝑎1𝑠

and 𝑏1𝑠

stand forthe longitudinal and lateral flapping angle of main rotor,respectively; the forces 𝑇

𝑀and 𝑇

𝑇and the moments 𝑄

𝑀and

𝑄𝑇can be calculated as [12, 13].Up to now, we have constructed the nonlinear dynamics

model of the RF-MJM system.

3. Analysis of DynamicsModel and Linearization

3.1. Analysis of the Dynamics Model. In order to understandthe coupling between the RFR and manipulator clearly, werewrite the system model (28) as follows:

[𝐻𝑏(Φ,Θ) 𝐻

𝑏𝑚(Φ,Θ)

𝐻𝑇𝑏𝑚(Φ,Θ) 𝐻

𝑚 (Φ,Θ)][

��𝑏

Θ] + [

[

𝐶𝑏(V0, 𝜔0,Θ,Φ, Φ)

𝐶𝑚(V0, 𝜔0,Θ,Φ, Φ)

]

]

+ [𝐺𝑏(𝑋,Θ,Φ)

𝐺𝑏(Θ,Φ)

] = [𝐹𝑝

𝜏𝑚

] .

(35)

From (35), it can be easily seen that the coupling between theRFR and the manipulator appears in all the terms except forthe exerting force/moment. That means the RF-MJM systemmodel is more complicated than the RFR system model asshown in [11]. These can be summarized as follows.

(1) Compared to the RFR model, there are some newterms in the system (28), such as𝐶

𝑏V.These new termsmake the system model more complicated, and theresult is that some control algorithm that has beenshown to be fit for RFR system cannot be used directlyin the RF-MJM system. For example, in reference[11], a RFR system is proved to be approximatefeedback linearizable. This, unfortunately, cannot beimplemented in the RF-MJM system.

(2) The system structure of the RF-MJM is of greatcomplication compared to the RFR system. This canbe easily seen through the preceding system equa-tions. Again, the reason is because of the couplingbetween the RFR and the manipulator. The higher


complication results in heavier nonlinearity whichmakes the controller design of the RF-MJM systemex-traordinarily challenging.

3.2. Linearization and LQR Controller Design. In the abovesection, we have obtained the detailed mathematical modelof the RF-MJM system, which can be easily computed andsimulated through using symbolic computation toolbox ofMATLAB. However, this kind of controller is difficult to beused due to the high complexity and nonlinearities. Thus, inthis section, we will try to find the linearizedmodel of the RF-MJM system and analyze the influence of the parameters onthe system parameters.

The linearization can be implemented through the follow-ing steps: firstly, search the trim point of the RF-MJM system;secondly, compute the derivatives of the system model withrespect to the state and input to obtain the system matrix ofthe desired linear model. With the linearized model, somelinear controller design strategies, for example, the LQRcontroller, can be used to stabilize the original nonlinearsystem. In the following content of this section, takingone-joint RF-MJM system as an example, the linearizationand the LQR control design will be conducted and systemperformance will be analyzed.

A trim point, also known as an equilibrium point, of anonlinear system is a point in the state space of a dynamicsystem, and at this point, the derivatives of the states withrespect to time are precisely zeros.

The state vector of the RF-MJM can be denoted as

[𝑥 𝑦 𝑧 𝜙 𝜃 𝜓 𝑢 V 𝑤 𝑝 𝑞 𝑟 Θ Θ] (36)

and the input vector is

[𝑎1𝑠 𝑏1𝑠 𝜃𝑀 𝜃𝑇𝜏Θ] , (37)

where 𝑎1𝑠and 𝑏

1𝑠are the cyclic pitch angle of the main rotor,

respectively; 𝜃𝑀

and 𝜃𝑇are the collective pitch angle of the

main rotor and the tilt rotor, respectively.Based on the definition of trim point, all the velocity state

should be set to zeros; that is,

𝑢 = V = 𝑤 = 𝑝 = 𝑞 = 𝑟 = Θ = 0. (38)

Also, all the derivatives of the states with respect to timeshould be zeros. Under condition (38), this is equivalent to

[��𝑏

Θ] = 𝐻

−1([𝐹𝑝

𝜏𝑚

] + [𝐽𝑇𝑏

𝐽𝑇𝑚

]𝐹𝑒− [

𝐶𝑏

𝐶𝑚

] − [𝐺𝑏

𝐺𝑏

]) = 07×1.

(39)

The right-hand side of (31) is only related to 𝜙, 𝜃, 𝜓, Θ,and input vector, so we have 9 free variables and 7 equalities.Furthermore, if we define 𝜏

Θ= 0 and𝜓 = 0, we will have only

7 free variables. Thus, the trim point can be obtained directlyby solving the nonlinear equalities (39), which can be easilyconducted using some searching function in MATLAB.

In order to evaluate the influence of the mass of themanipulator on the whole system, we list out the trim point

Table 1: Parameters of RF-MJM system.

Parameter Describe Unit𝑚0= 9.5 The mass of rotor-flying robot (RFR) kg

𝑚1= 2.5 The mass of manipulator kg

𝐼0𝑥𝑥

= 0.1634 Moment of inertia of RFR kgm2

𝐼0𝑦𝑦


𝐼0𝑧𝑧


𝐼1𝑥𝑥

= 0.1399 Moment of inertia of manipulator kgm2

𝐼1𝑦𝑦


𝐼1𝑧𝑧


𝑙0= 0.3 The length from the centroid of RFR m

𝑙1= 0.4 The length of half of the first link m

𝜙 = 0.0769 Roll angle rad𝜃 = 0.0211 Pitch angle rad𝜓 = any value Yaw angle rad𝜃1= 0.0211 Joint movement angle rad

𝐹0= [0 0 0]

𝑇

External force N

𝑇0= [0 0 0]

𝑇

External torque Nm

of the linearization system with different manipulator masses(the parameters of the system are listed out in Table 1, and thetrim point is in Table 2 in the next section).

The linearization system model is

Δ�� = 𝐴Δ𝑋 + 𝐵Δ𝑢, (40)

whereΔ𝑋 = 𝑋 − 𝑋trim

𝐴 =

[[[[[[[

[

03×3

𝐸303×3

03×3

03×1

03×1

𝐴1

𝐴2

𝐴3

𝐴4

𝐴5

𝐴6

03×3

03×3

03×3

𝐸 03×1

03×1

𝐴7

𝐴8

𝐴9𝐴10

𝐴11

𝐴12

01×3

01×3

01×3

01×3

0 1

𝐴13

𝐴14

𝐴15

𝐴16

𝐴17

𝐴18

]]]]]]]

]

𝐵 = [05×3

𝐵𝑇105×3

𝐵𝑇205×1

𝐵𝑇3]𝑇

.

(41)

Furthermore, in order to analyze the performance ofsystem (40), the eigenvalues of𝐴matrix are given in Figure 5.From it, we can get the following results.

(1) Thewhole system is static-instable since it has positiveeigenvalues.

(2) With increase of the manipulator mass, the distribu-tion of eigenvalues will be more diverging.

3.3. LQR Controller Design. Next, we will design the fullstate-feedback linear quadratic regulation (LQR) controllerfor the RFM system. In the state-feedback version of theLQR problem [14], we assume that the whole state 𝑥 can bemeasured and therefore it is available to control. Solution tothe optimal state-feedback LQR problem is to find 𝑢(𝑡) =−𝐾𝑥(𝑡) that minimizes

𝐽LQR = ∫∞

0

(𝑥𝑇𝑄𝑥 + 𝑢

𝑇𝑅𝑢) 𝑑𝑡, (42)


0 2 4 6 8

0

0.5

1

1.5

Real

Imag

e−1

−0.5

−1.5−8 −6 −4 −2

m1 = 2.5

m1 = 2

m1 = 1.5

m1 = 0.5

m1 = 1

Figure 5: Eigenvalue distribution of 𝐴matrix with different manipulator masses.

Table 2: Trim points corresponding to different masses of manipulator.

Trim point 𝑚1= 2.5 kg 𝑚

1= 2 kg 𝑚

1= 1.5 kg 𝑚

1= 1 kg 𝑚

1= 0.5 kg

𝜙 0.0769 0.0732 0.0697 0.0662 0.0626𝜃 0.0211 0.0204 0.0197 0.0190 0.0183𝜃1

0.0211 0.0205 0.0198 0.0191 0.0183𝜃𝑀

0.0436 0.0409 0.0381 0.0354 0.0326𝜃𝑇

−0.1282 −0.1245 −0.1207 −0.1169 −0.1130𝑎1𝑠

0.0211 0.0204 0.0197 0.0190 0.0183𝑏1𝑠

0.0195 0.0169 0.0145 0.0123 0.0103

where 𝐾 is given by 𝐾 = 𝑅−1𝐵𝑇𝑃 and 𝑃 is found by solvingsome continuous time algebraic Riccati equations. So we caneasily get the eigenvalues of the open loop system and theclosed loop systemby thematrices𝐴 and𝐴−𝐵𝐾, respectively.

4. Simulations

In this section, simulations will be conducted using thepreceding nonlinear system model and the LQR controller.In the simulation, the manipulator’s mass is 2.5 kg, and theother parameters are given in Table 1.

And the trim point of the whole system is listed out inTable 2.

With these parameters, the system matrices 𝐴 and 𝐵 areas follows:

𝐴 =

[[[[[[[

[

03×3

𝐸 03×3

03×3

03×1

03×1

03×3

03×3

𝐴303×3

𝐴503×1

03×3

03×3

03×3

𝐸 03×1

03×1

03×3

03×3

𝐴903×3

𝐴11

03×1

01×3

01×3

01×3

01×3

0 1

01×3

01×3

𝐴15

01×3

𝐴17

0

]]]]]]]

]

𝐵 = [014×3

𝐵𝑇

1014×3

𝐵𝑇

2014×1

𝐵𝑇

3]𝑇

,

(43)

where

𝐴3=[[

[

0.0015 9.7636 0.9945

−9.8000 −0.0005 0.2102

0.0001 0.0005 0.0014

]]

]

;

𝐴9=[[

[

−0.0087 −0.0083 0.1834

−0.0130 0.0255 −0.8587

−0.2013 −0.2456 −0.0635

]]

]

;

𝐴5=[[

[

3.5861

−0.0077

0.0310

]]

]

; 𝐴11= [

[

−0.0547

−24.4857

−0.8988

]

]

;

𝐴15= [0.0110 −0.0924 2.6951] ;

𝐴17= 40.1487;

𝐵𝑇

1=

[[[[[

[

−0.5451 −7.9183 −147.7390

−1.2434 −9.3478 0.6158

−21.7522 0.4534 −0.2015

−0.5829 7.2377 −0.7650

0.2990 −0.0006 0.0009

]]]]]

]

;


0 5 10 15 20

00.05

0.10.15

0.20.25

0.30.35

Time (s)

Posit

ion

(m)

NorthEastHeight

−0.1

−0.05

(a)

5 10 15 20Time (s)

Line

ar v

eloci

ty (m

/s)

00.05

0.10.15

0.2

0.30.25

−0.2

−0.1

−0.15

−0.05

0

VnorthVeastVheight

(b)

0 5 10 15 20

0

0.02

0.04

0.06

0.08

Time (s)

Attit

ude (

rad)

−0.02

−0.04

𝜙

𝜓𝜃

(c)

0 5 10 15 20

00.05

0.10.15

0.20.25

Time (s)

Ang

ular

velo

city

(rad

/s)

−0.2

−0.1

−0.15

−0.05

𝜔1𝜔2𝜔3

(d)

0 5 10 15 20

00.020.04

Time (s)

−0.02

−0.04

−0.06

−0.08

−0.1

−0.12Join

t ang

le an

d ve

loci

ty (r

ad) (

rad/

s)

𝜃dot

𝜃1

(e)

0 5 10 15 20

0

0.05

0.1

Time (s)

Con

trol i

nput

−0.05

−0.15

−0.1

𝜃M𝜃Ta1

b1𝜏

(f)

Figure 6: States and inputs profile under LQR controller (46).

𝐵𝑇

2=

[[[[[[

[

16.7653 −2.5282 168.3611

−32.8659 14.0449 125.4196

20.9082 178.3726 4.0958

−121.2649 11.0285 −0.2925

0.0011 0.6932 −0.0534

]]]]]]

]

;

𝐵𝑇

3= [−3.9371 −9.0113 −99.1817 −4.2009 3.4137] .

(44)

From these equations, it can be seen that the couplingbetween the manipulator and RFR, denoted by 𝐴

5, 𝐴

11, 𝐴

15,


0 5 10 15 20 25 30 35 40 45

00.01

Time (s)

−0.02

−0.01Ph

(m)

Ph1 Ph2

0 5 10 15 20 25 30 35 40 45

0

0.1

−0.1

Pea

st(m

)

Peast1 Peast2

0 5 10 15 20 25 30 35 40 45

0

0.5

−0.5Pno

rth

(m)

Pnorth1 Pnorth2

(a)

Time (s)

𝜙1 𝜙2

0 5 10 15 20 25 30 35 40 45

0 5 10 15 20 25 30 35 40 45

0 5 10 15 20 25 30 35 40 45

𝜃1 𝜃2

0

0.1

0.080.075

0.0650.07

−0.1

00.02

−0.04

−0.02

𝜃(r

ad)

𝜙(r

ad)

𝜓(r

ad)

𝜓1 𝜓2

(b)

0 5 10 15 20 25 30 35 40 45

0 5 10 15 20 25 30 35 40 45

0 5 10 15 20 25 30 35 40 45

0

0.5

−0.5

Vno

rth

(m)

Vea

st(m

)

0

0.040.02

0

0.05

−0.02

−0.05

Vh1 Vh2

Veast1 Veast2

Vnorth1 Vnorth2

Vh

(m)

Time (s)

(c)

0 5 10 15 20 25 30 35 40 45

0 5 10 15 20 25 30 35 40 45

0 5 10 15 20 25 30 35 40 45

00.020.04

−0.02

Time (s)

0

0.2

−0.2

0

0.5

−0.5

𝜔2

(rad

/s)

𝜔3

(rad

/s)

𝜔1

(rad

/s)

𝜔1-1

𝜔2-1 𝜔2-2

𝜔3-2𝜔3-1

𝜔1-2

(d)

0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 30 35 40 450.01

0.0150.02

0.025

0.0350.03 0

0.05

−0.15

−0.05

−0.1

Time (s) Time (s)

𝜃

𝜃

𝜃1

(rad

)

𝜃1

dot

(rad

/s)

𝜃1dot

𝜃1dot1-1 -1

-21-2

(e)

Figure 7: Continued.


0 5 10 15 20 25 30 35 40 45

0 5 10 15 20 25 30 35 40 45

0 5 10 15 20 25 30 35 40 45

0 5 10 15 20 25 30 35 40 45

0 5 10 15 20 25 30 35 40 45

0

0

0.01

0.010.02

0.020.040.06

0.050.1

0.03

𝜏 (N

m)

Time (s)

𝜃M

(rad

)𝜃T

(rad

)a 1

(rad

)b 1

(rad

)

−0.01

−0.12−0.14−0.16

𝜃 𝜃

𝜃 𝜃

b1-1 b1-2

a1-2a1-1

T-1 T-2

M-2M-1

𝜏-1 𝜏-2

(f)

Figure 7: States response with disturbance (49) (blue and solid) and (50) (red and dashed)∗; the subscript -1 means that the results are withthe disturbance (49) and the subscript -2 means that the results are with the disturbance (50).

and 𝐴17

is heavy. Using the following QR parameters in theLQR controller,𝑄 = diag (1, 1, 1, 1, 1, 1, 10, 10, 10, 10, 10, 10, 10, 10) ;

𝑅 = diag (1000, 1000, 1000, 1000, 1000) ,(45)

the LQR feedback control law is designed as𝑢 = 𝐾Δ𝑥

𝐾 = [𝐾1 𝐾2 𝐾3 𝐾4 𝐾5] ,(46)

where

𝐾1=

[[[[[[[

[

0.0097 −0.0037 −0.0226

0.0119 0.0058 0.0210

−0.0232 −0.0014 0.0013

−0.0018 0.0308 −0.0066

0.0149 −0.0006 −0.0008

]]]]]]]

]

𝐾2=

[[[[[[[

[

0.0200 −0.0071 −0.0291

0.0238 0.0116 0.0287

−0.0544 −0.0019 0.0031

−0.0042 0.0575 −0.0079

0.0308 −0.0013 −0.0016

]]]]]]]

]

𝐾3=

[[[[[[[

[

0.0502 0.1489 0.1317

−0.0816 0.1670 0.1279

0.0053 −0.4953 −0.3462

−0.3754 −0.0384 −0.0293

0.0104 0.2401 0.0991

]]]]]]]

]

𝐾4=

[[[[[[[

[

0.0169 0.0637 0.0751

−0.0226 0.0663 0.0827

−0.0015 −0.2717 −0.0288

−0.1208 −0.0214 −0.0058

0.0040 0.1085 0.0073

]]]]]]]

]

𝐾5=

[[[[[[[

[

0.6994 0.1310

0.5973 0.1125

−4.2835 −0.8102

−0.3735 −0.0691

1.1293 0.2157

]]]]]]]

]

.

(47)

With the controller (46), the system can be stabilizedwith acceptable performance near the trim point, and thesimulation results with initial state (0.1, 0, −0.1, 0, 0, 0, 0.07,0.04, 0, 0, 0, 0.03, 0) are shown in Figure 6.

From Figure 6, it can be seen that a linear LQR controllercan stabilize the whole system near the trim point. However,the stabilizing region of LQR is very limited; we have testedthat only when the attitude of the whole system satisfies thefollowing conditions (all the initial velocities are set to zeros),the LQR control is effective (46):

−0.1389 ≤ 𝜃 ≤ 0.3911

−0.6731 ≤ 𝜙 ≤ 0.5469

−0.0619 ≤ 𝜃1≤ 0.0611.

(48)

Now in the next simulation, two periodic sinusoidalsignals, as disturbances with different frequencies, are added


0 2 4 6 8 10 12 14 16 18 20

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (s)

Error 1Error 2Error 3

−0.4

−0.2

Ang

ular

acce

lera

tion

erro

rs (r

ad/s

2 )

(a)

0 2 4 6 8 10 12 14 16 18 20

0

0.5

1

1.5

2

2.5

Time (s)

−2

−1.5

−0.5

−1

Error 4Error 5Error 6

Line

ar ac

cele

ratio

n er

rors

(m/s2)

(b)

Figure 8: Decoupling between RFR and manipulator.

to the input of themanipulator to test motion influence of themanipulator on the whole system:

𝑑1= 0.01 sin (0.5𝜋𝑡) (49)

𝑑2= 0.01 sin (1.0𝜋𝑡) . (50)

The results are as in Figure 7.Simultaneously, in order to test the coupling between

the RFR and the manipulator, the linear accelerations andthe angular accelerations of the new RF-MJM system arecompared to the helicopter system with the same parametersas in Table 1; that is,

[

[

error 1error 2error 3

]

]

= [

[

𝑎RF−MJM,𝑥𝑎RF−MJM,𝑦𝑎RF−MJM,𝑧

]

]

− [

[

𝑎RFR,𝑥𝑎RFR,𝑦𝑎RFR,𝑧

]

]

;

[

[

error 4error 5error 6

]

]

= [

[

��RF−MJM,𝑥��RF−MJM,𝑦��RF−MJM,𝑧

]

]

− [

[

��RFR,𝑥��RFR,𝑦��RFR,𝑧

]

]

.

(51)

The results are given in Figure 8, which presents the extraforce andmoment exerted on the RFR due to themanipulatorand its motion.

5. Conclusions

In this paper, the detailed nonlinear dynamics model ofa rotor-flying multijoint (RF-MJM) system is constructedthrough using Euler-Lagrange method. Compared to therotor-flying vehicle system, the model nonlinearities and

complexity of the new RF-MJM are analyzed in detail. More-over, linear analysis is conducted with respect to the con-structed nonlinear model near its trim point, and the influ-ence of the manipulator mass on the system’s local per-formance is researched. Furthermore, LQR controller isdesigned based on the linearized system model. Finally,simulation results show that (1) a linear LQR controller isable to stabilize the system near steady state and presentsacceptable performance; however, (2) the stabilization regionof LQR controller is very limited, and the performance ofLQR controller is sensitive to the external disturbance. Thus,in the future work, nonlinear and robust control scheme willbe researched to overcome the disadvantages of the linearcontroller.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China (no. 61305120 and 61035005).

References

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[3] G. Cai, B. M. Chen, and T. H. Lee, “An overview on develop-ment of miniature unmanned rotorcraft systems,” Frontiers ofElectrical and Electronic Engineering in China, vol. 5, no. 1, pp.1–14, 2010.

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[8] D. Mellinger, M. Shomin, N. Michael, and V. Kumar, “Coop-erative grasping and transport using multiple quadrotors,” inDistributedAutonomous Robotic Systems, pp. 545–558, Springer,2013.

[9] P. E. Pounds, D. R. Bersak, and A. M. Dollar, “Grasping fromthe air: hovering capture and load stability,” in Proceedings ofthe IEEE International Conference on Robotics and Automation(ICRA ’11), pp. 2491–2498, IEEE, 2011.

[10] X. Y. Xu and T. Kanade, Space Robotics: Dynamics and Control,Springer, 1993.

[11] Y. Q. He and J. D. Han, “Acceleration-feedback-enhancedrobust control of an unmanned helicopter,” Journal of Guidance,Control, and Dynamics, vol. 33, no. 4, pp. 1236–1250, 2010.

[12] A. R. Bramwell, D. Balmford, and G. Done, Bramwell’s Heli-copter dynamics, Butterworth-Heinemann, 2001.

[13] D. H. Shim, Hierarchical control system synthesis for rotorcraft-based unmanned aerial vehicles [Ph.D. thesis], Berkeley, Calif,USA, 2000.

[14] J. Hespanha, “Undergraduate lecture notes on lqg/lqr controllerdesign,” in Apuntes Para Licenciatura. UCBS, pp. 1–37, Univer-sity of California, Santa Barbara, Calif, USA, 2007.

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