Open access peer-reviewed chapter

Nonlinear Dynamics and Control of Aerial Robots

By Mahmut Reyhanoglu and Muhammad Rehan

Submitted: October 18th 2016Reviewed: May 8th 2017Published: September 6th 2017

DOI: 10.5772/intechopen.69641

Downloaded: 530

Abstract

Aerial robotics is one of the fastest growing industry and has a number of evolving applications. Higher agility make aerial robots ideal candidate for applications like rescue missions especially in difficult to access areas. This chapter first derives the complete nonlinear dynamics of an aerial robot consisting of a quadcopter with a two-link robot manipulator. Precise control of such an aerial robot is a challenging task due to the fact that the translational and rotational dynamics of the quadcopter are strongly coupled with the dynamics of the manipulator. We extend our previous results on the control of quadrotor UAVs to the control of aerial robots. In particular, we design a backstepping and Lyapunov-based nonlinear feedback control law that achieves point-to-point control of the areal robot. The effectiveness of this feedback control law is illustrated through a simulation example.

Keywords

  • quadcopter
  • robot manipulator
  • backstepping method
  • nonlinear control

1. Introduction

The recent surge of interest in applications involving unmanned aerial vehicles (UAVs) has inspired several research efforts in UAV dynamic modeling and control. In particular, nonlinear control of fixed-wing UAVs has attracted considerable research efforts during recent years both for civilian and military purposes. The control approaches developed for fixed-wing UAVs include gain scheduling, model predictive control, backstepping, sliding mode, nested saturation, fuzzy control, H control, dynamic inversion based control, model reference adaptive control, and model based fault tolerant control [112].

While control applications involving fixed-wing UAVs have been widely investigated in recent literature, quadrotor UAV (quadcopter) control applications are growing in popularity due to their maneuverability and versatility. Quadcopters offer practical advantages over fixed-wing UAVs in military and civilian applications involving search and rescue, area mapping, and surveillance. However, actuator constraints, sensor limitations, and the high degree of nonlinearity and uncertainty inherent in the system dynamics present specific challenges in control system design for quadcopters. Linear control approaches, such as PID, LQR or LQG are popularly utilized to address the quadcopter control problem [1315]. Although linear control methods have been shown to perform well in their respective quadcopter control tasks, the effectiveness of linear control methods can only be guaranteed over a limited range of operating conditions. The highly agile nature of quadcopters necessitates the development of control methods that can be applied over a wide range of time-varying, uncertain, and potentially adversarial operating conditions. To achieve reliable quadcopter control over a wider operational envelope, several nonlinear control methods have recently been presented. Popular nonlinear control methods for quadrotor systems include backstepping, feedback linearization, dynamic inversion, adaptive control, Lyapunov-based robust control, fuzzy-model approach, and sliding mode control [1619]. In Ref. [16], a passivity-based quaternion feedback control strategy is presented for a hover system (quadrotor UAV test bed), which achieves asymptotic attitude regulation. The proposed control design incorporates the input voltage constraints inherent in practical UAV systems. A rigorous Lyapunov-based analysis is provided to prove asymptotic regulation of the hover system attitude to a desired set point. In Ref. [17], a sliding mode control (SMC) strategy is presented for a quadrotor-based hover system, which achieves asymptotic attitude regulation in the presence of electrical and physical constraints. A sliding mode observer is employed to estimate the angular velocities. In addition, the proposed control design incorporates the input voltage constraints inherent in practical systems. A rigorous Lyapunov-based analysis is provided to prove asymptotic regulation of the hover system attitude to a desired set point.

Aerial robotics is one of the fastest growing industry and has a number of evolving applications. Higher agility make aerial robots ideal candidate for applications like rescue missions especially in difficult to access areas. Furthermore, swarm robotics (multiple robot working together) is another exciting application of the aerial robotics, for example coordinated assembly at higher altitudes. These robots can behave like individuals working in a group without centralized control. Researchers have developed intelligent control algorithms for the swarms after deep study of animal behavior in herds, bird flocks and fish schools. In some applications, for an aerial robot, linear control theory works well but these control techniques are effective in a limited operating regions. Moreover, the motion of arm induces disturbances to the quadcopter dynamics so the linear controllers lose their effectiveness during operation and sometime the closed loop system becomes unstable. In order to accomplish complex missions in presence of uncertainties in the environment, to achieve better maneuverability and precise 3D position and attitude control, nonlinear control techniques have been found effective [2027], In Ref. [20] a set of nonlinear control laws have been proposed for aerial manipulator that provide asymptotic attitude and position tracking. Backstepping-based nonlinear control scheme for automatic trajectory tracking for aerial manipulators has been proposed in Refs. [22, 24].

In this chapter, we extend our results on the control of quadcopters to the control of aerial robots. We derive the complete nonlinear dynamics of an aerial robot consisting of a quadcopter with a two-link robot manipulator. Precise control of such a robot is a challenging task because attitude and position dynamics of the quadcopter are strongly coupled with the dynamics of the manipulator. We develop nonlinear control laws that ensure the control of position and attitude of the aerial robot. Simulation results are included to demonstrate the effectiveness of the control laws.

2. Mathematical model

This section formulates the dynamics of an aerial robot consisting of a quadcopter with a two-link robot manipulator. The quadcopter is represented as a base body and the links as internal bodies. The equations of motion are expressed in terms of the three dimensional translational velocity vector, the attitude, the angular velocity, and the internal (shape) coordinates representing the configuration of the two links.

2.1. Multibody vehicle dynamics

Following the development in [28], let vR3, ωR3, and ηS1×S1denote the base body translational velocity vector, the base body angular velocity vector, and the vector of internal coordinates, respectively. In these variables, the kinetic energy has the form T=T(v,ω,η,η˙), which is SE(3)-invariant in the sense that it does not depend on the base body position and attitude. The equations of motion of the quadrotor with internal dynamics are shown to be given by:

ddtTν+ω^Tν=Ft,E1
ddtTω+ω^Tω+ν^Tν=τr,E2
ddtTη˙Tη=τs,E3

where FtR3, τrR3denote the vectors of generalized control forces and generalized control torques, respectively, that act on the base body, and τsR3is the vector of joint torques. For a given vector a = [a1 a2 a3]TR3, the skew-symmetric matrix a^defines the corresponding cross-product operation a×, given by

a^=[0a3a2a30a1a2a10].E4

Note that Eqs. (1) and (2) are identical to Kirchhoff’s equations [29], which can also be expressed in the form of Euler-Poincaré equations.

2.2. Nonlinear equations of motion

Consider an aerial robot that consists of a quadcopter with a two DOF manipulator arm moving in a three-dimensional space as shown in Figure 1, where p = [x y z]T denotes the inertial position of the center of mass of the quadcopter. Denote by xyz axes the inertial frame F1 and by xB yB zB axes the body-fixed frame FB with the origin at the CM of the quadcopter. Let R denote the attitude matrix of the quadcopter and (v, ω) be the translational and angular velocities of the CM of the quadrotor in FB. Then, the translational and rotational kinematics can be expressed as

Figure 1.

Model of a quadcopter with a robotic arm.

p˙=Rv,E5
R˙=Rω^.E6

The quadcopter consists of four propellers connected to a rigid frame. Each propeller is mounted on the frame at a distance l from the origin. The quadcopter has a mass m and inertia matrix J defined with respect to the axes of rotation. Due to symmetry of the system, J is diagonal, that is, J = diag{Jxx, Jyy, Jzz}. We refer to rotation about the xB-axis as roll, rotation about the yB-axis as pitch, and rotation about the zB-axis as yaw. The propellers generate lift forces

Fi=bΩi2=bKν2Vi2,E7

where Ωi, Vi denote, respectively, the angular rate and input voltage for propeller i, and b is the thrust coefficient. The total thrust is given by

Fp=i=14Fie3=bKv2(V12+V22+V32+V42)e3,E8

where e3 = [0 0 1]TR3is the third standard basis vector.

Note that, as shown in Figure 2, propellers 1 and 3 rotate clockwise, and propellers 2 and 4 rotate counter-clockwise. By balancing the torque between opposing propellers, the roll and pitch angle can be controlled. Since all four propellers generate a net torque about the yaw axis, the yaw angle can be controlled by balancing the torque generated by clockwise and counter-clockwise rotating propellers.

Figure 2.

Model of the quadcopter.

The arm is attached at the CM of the quadcopter and it can only move in xz-plane of the body-fixed frame FB. The physical constants are the quadcopter mass m, the link masses mi, i = 1,2, and the payload mass mp. Let lcidenote the distance from joint i to the CM of link i and li be the length of link i. The position vectors for the CM of the links and the payload with respect to the CM of the base body in FB can be written as

ρ1=[lc1cosθ10lc1sinθ1]T,E9
ρ2=[l1cosθ1+lc2cosθ20l1sinθ1+lc2sinθ2]T,E10
ρp=[l1cosθ1+l2cosθ20l1sinθ1+l2sinθ2]T.E11

Let Ft = Fg +Fp, where Fg and Fp denote, respectively, the gravitational force acting on aerial robot and the total thrust generated by the four propellers. Also let τr = τg + τp, where τg and τp are the torque acting on the aerial robot due to gravity and the torque generated by the propellers, respectively.

Clearly, Fg and τg can be computed as

Fg=mtgRTe3,E12
τg=g[m1ρ1+m2ρ2+mpρp]×RTe3,E13

where mt = m + m1 + m2 + mp.

The generalized torque vector τp (expressed in the body frame) comprises the following components:

  • Propellers 2 and 4 generate a moment l(F4F2)=blKν2(V42V22)about the roll axis.

  • Propellers 1 and 3 generate a moment l(F3F1)=blKν2(V32V12)about the pitch axis.

  • The sum of all torques about z-axis is dKν2(V12V22+V32V42)and causes a yaw moment.

  • The rotation of the propellers causes the gyroscopic effect JrKνωy(V1V2+V3V4)about the roll-axis and JrKνωx(V1V2+V3V4)about the pitch-axis.

Here d denotes the drag coefficient, l is the distance from the pivot to the motor, and Jr is the rotor inertia. Combined, the generalized torque τp can be expressed as

τp=[blKν2(V42V22)+JrωyΩrblKν2(V32V12)JrωxΩrdKν2(V12V22+V32V42)],E14

where Ωr:=Kν(V1V2+V3V4)is the overall residual angular speed of the propellers.

Let η=[θ1θ2]Tdenote the shape variables. Then, the linear and angular velocities of each link, and the linear velocity of the payload can be expressed in FB as

νi=ν+ω^ρi+ρiηη˙=νρ^iω+ρiηη˙,i=1,2,E15
ωi=ω+θ˙ie2=ω+Ci(η)η˙,i=1,2,E16
νp=ν+ω^ρp+ρpηη˙=νρ^pω+ρpηη˙,E17

where e2 = [0 1 0]TR3is the second standard basis vector. The total kinetic energy can now be expressed as

T(ν,ω,η,η˙)=12mνTν+12ωTJω+12i=12(miνiTνi+ωiTJiωi)+12mpνp2,E18

where Ji=RiTJ¯iRiis the inertia matrix of the ith link with respect the body frame FB, Ri is the rotation matrix of the ith link, which is given by

Ri=[cosθi0sinθi010sinθi0cosθi],i=1,2,E19

and J¯idenotes the inertia matrix of the ith link with respect to xiyizi-axes attached to the link. Assuming the two links are made up of homogeneous rods, J¯ican be expressed as

J¯i=[0000112mili2000112mili2].E20

Applying Kirchhoff’s equations (1) and (2), the complete nonlinear equations of motion can be obtained as

[MKBtKTJ¯BrBtTBrTm¯][v˙w˙η¨]=[Ftτrτs][w^MK˙+w^KB˙t+w^BtK˙T+w^KT+ν^MJ¯˙+w^J¯+ν^KB˙a+w^Ba+ν^BtB˙tTB˙aTm¯˙][vwη˙]+[00Lη],E21

where

M=mtI3×3,E22
J¯=J+m1ρ^1Tρ^1+m2ρ^2Tρ^2+mpρ^pTρ^p+J1+J2,E23
m¯=m1ρ1ηTρ1η+m2ρ2ηTρ2η+mpρpηTρpη+C1TJ1C1+C2TJ2C2,E24
K=m1ρ^1m2ρ^2mpρ^p,E25
Bt=m1ρ1η+m2ρ2η+mpρpη,E26
Br=m1ρ^1ρ1η+m2ρ^2ρ2η+mpρ^pρpη+J1C1+J2C2.E27

Complete description of the above coefficient matrices are given in the appendix. The objective is to simultaneously control the 6 DOF motion of the quadcopter and the 2 DOF internal dynamics of the robot arm using only 4 propellers and 2 joint torque motors. In this regard, equations of motion given by (21) represents an interesting example of underactuated mechanical systems. In our previous research [3032], we have developed theoretical controllability and stabilizability results for a large class of underactuated mechanical systems using tools from nonlinear control theory. We have also developed effective nonlinear control design methodologies [32] that we applied to several examples of underactuated mechanical systems, including underactuated space vehicles [33].

3. Nonlinear control design

The translational and rotational dynamics of the quadcopter are coupled with the dynamics of its robotic arm; this makes controller design very complicated. The equations of motion in component form are given by

Mν˙+Kω˙+Btη¨=Ftω^Mν(K˙+ω^K)ω(B˙t+ω^Bt)η˙,E28
KTν˙+J¯ω˙+Baη¨=τr(K˙T+ω^KT+ν^M)ν(J¯˙+ω^J¯+ν^K)ω(B˙a+ω^Ba+ν^Bt)η˙,E29
BtTν˙+BaTω˙+m¯η¨=τsB˙tTνB˙aTωm¯˙η˙+Lη.E30

Eq. (28) can be rewritten as

Mν˙=Ftω^Mν+Fd,E31

where

Fd=Kω˙Btη¨(K˙+ω^K)ω(B˙t+w^Bt)η˙.E32

Eq. (31) can be simplified as

[x¨y¨z¨]=[00g]+Re^3u1+F¯d.E33

where F¯d=Fd/mtand

u1=bKv2(V12+V22+V32+V42)/mt,E34

Similarly, Eq. (29) can be rewritten as

J¯ω˙=τr(J¯˙+ω^J¯)ω+τd,E35

where

τd=KTv˙Baη¨(K˙T+ω^KT+ν^M)νν^Kω(B˙a+ω^Ba+ν^Bt)η˙.E36

Eq. (35) can be simplified as

ω˙=J¯1τrJ¯1(J¯˙+ω^J¯)ω+τ¯d,E37

where τ¯d=J¯1τd.

Ignoring F¯dand τ¯din Eqs. (33) and (37), equations of motion can be expressed as

x¨=(cosφsinθcosΨ+sinφsinΨ)u1,E38
y¨=(cosφsinθsinΨsinφcosΨ)u1,E39
z¨=g+(cosφcosθ)u1,E40
[ω˙xω˙yω˙z]=J¯1(τg+[JrωyΩrJrωxΩr0](J¯˙+ω^J¯)ω)+[u2u3u4],E41

where

[u2u3u4]=J¯1[blKν2(V42V22)blKν2(V32V12)blKν2(V12V22+V32V42)].E42

We now design a nonlinear controller based on integrator backstepping. If φ, θ and Ψ are small (sin θθ and cos θθ), then ω[φ˙θ˙Ψ˙]Tand ω˙[φ¨θ¨Ψ¨]T, and hence the equation of motions can be simplified as

x¨=θu1,E43
y¨=φu1,E44
z¨=g+u1,E45
φ¨=f1(φ,θ,Ψ)+u2,E46
θ¨=f2(φ,θ,Ψ)+u3,E47
Ψ¨=f3(φ,θ,Ψ)+u4,E48

where

[f1f2f3]=J¯1(τg+[Jrθ˙ΩrJrφ˙Ωr0](J¯˙+ω^J¯)[φ˙θ˙Ψ˙]).E49

3.1. Controller design

In this section a nonlinear controller is designed to stabilize the system (43)–(48) to the desired equilibrium configuration (x,y,z,φ,θ,Ψ)=(xd,yd,zd,φd,θd,Ψd).

We choose u1 as

u1=g|zzd|asign(zzd)|z˙|bsign(z˙),E50

where b ∈ (0, 1), a > b / (2 − b), i = 1,2, are controller parameters. The feedback law (50) controls the quadcopter z-dynamics to (z,z˙)=(zd,0)in finite time [34] so that u1g in finite time.

After reaching the desired altitude, Eqs. (43) and (44) take the following form:

x¨=gθ,E51
y¨=gφ.E52

We now apply a backstepping method to design the controls u2 and u3 to stabilize the system to the equilibrium at (x,y,φ,θ)=(xd,yd,φd,θd).

Assume that θ and φ are virtual inputs for the x and y subsystems, respectively. Stabilizing feedback functions for the x-subsystem is given by

θ=k1(xxd)k2x˙,E53
φ=k3(yyd)+k4y˙,E54

where ki > 0, i = 1,…,4, so that

x¨+gk2x˙+gk1(xxd)=0,E55
y¨+gk3y˙+gk4(yyd)=0.E56

Define

y1=θ+k1(xxd)+k2x˙,E57
y2=φk3(yyd)k4y˙,E58

and consider the y1 and y2 dynamics given by

y˙1=θ˙+k1x˙+k2x¨=θ˙+k1x˙+k2gθ,E59
y˙2=φ˙k3y˙k4y¨=φ˙k3y˙+k4gφ.E60

Define the sliding variables (s1, s2)

s1=y˙1+α1y1,E61
s2=y˙2+α2y2,E62

where αi > 0, i = 1,2, which can be simplified as

s1=θ˙+(k2g+α1)θ+(k1+α1k2)x˙+α1k1(xxd),E63
s2=φ˙+(k4g+α2)φ(k3+α2k4)y˙α2k3(yyd).E64

The dynamics of sliding variables are found simply by taking time derivative of the sliding variables as

s˙1=θ¨+(k2g+α1)θ˙+(k1+α1k2)gθ+α1k1x˙,E65
s˙2=φ¨+(k4g+α2)φ˙+(k3+α2k4)gφα2k3y˙.E66

Substituting the expressions for φ¨and θ¨from (46) and (47), respectively, we obtain

s˙1=f2(φ,θ,Ψ)+u3+(k2g+α1)θ˙+(k1+α1k2)gθ+α1k1x˙,E67
s˙2=f1(φ,θ,Ψ)+u2+(k4g+α2)φ˙+(k3+α2k4)gφα2k3y˙.E68

We choose the inputs u2 and u3 as

u2=λ2sign(s2)f1(φ,θ,Ψ)(k4g+α2)φ˙(k3+α2k4)gφ+α2k3y˙,E69
u3=λ1sign(s1)f2(φ,θ,Ψ)(k2g+α1)θ˙(k1+α1k2)gθα1k1x˙,E70

so that the following closed-loop response for the sliding variables is obtained:

s˙1=λ1sign(s1),E71
s˙2=λ2sign(s2),E72

where we choose λ1 > 0 and λ2 > 0 large enough so that the terms Fd and τd are dominated by the sliding mode terms.

Now consider the Ψ-dynamics given by (48). The following control law stabilizes the Ψ-dynamics to (Ψ,Ψ˙)=(Ψd,0):

u4=k5(ΨΨd)k6ψ˙f3(φ,θ,Ψ),E73

where k5, k6 > 0.

The voltage inputs Vi, i = 1, …, 4, are determined by substituting the expressions for the virtual control inputs ui, i = 1, …, 4, into Eqs. (34) and (42).

Consider Eqs. (31) and (35), and ignore Fd and τd. Then we have

ν˙=M1FtM1w^Mν,E74
ω˙=J¯1τrJ¯1(J¯˙+ω^J¯)ω.E75

Eq. (30) can be rewritten as

η¨=m¯1(BtTν˙BaTw˙+τsB˙tTνB˙aTwm¯˙η˙+Lη),E76

which can be expressed in terms of Ft and τr as

η¨=m¯1(τsBtTFtmtBtTω^νBaTJ¯1(τr(J¯˙+ω^J¯)ω)B˙tTνB˙aTwm¯˙η˙+Lη).E77

In order to have exponential convergence of the shape variables η to the desired ηd we choose τs as

τs=BtTFtmt+BtTω^v+BaTJ¯1(τr(J¯˙+ω^J¯)ω)+B˙tTν+B˙aTw+m¯˙η˙Lηm¯(2λη˙+λ2(ηηd))E78

where λ > 0, so that

η¨+2λη˙+λ2(ηηd)=0.E79

4. Simulation

The controller developed in the previous sections is applied to the full nonlinear model of the aerial robot. The relevant parameter values of the system are listed in Table 1.

SymbolParameterValueUnit
KvTransformation constant54.945rad s V−1
JrRotor inertia6 × 10−5kg m2
JxxMOI about x axis0.0552kg m2
JyyMOI about y axis0.0552kg m2
JzzMOI about z axis0.1104kg m2
bThrust coefficient3.935139 × 10−6N V−1
dDrag coefficient1.192564 × 10−7Nm V−1
lDistance from pivot to motor0.1969m
mMass2.85kg
gAcceleration of gravity9.81ms−2
VMaximum input voltage10V
m1Mass of link 10.1kg
m2Mass of link 20.1kg
mpMass of the payload0.1kg
l1Length of link 10.5m
l2Length of link 20.5m

Table 1.

Parameters of the aerial robot.

A rest-to-rest motion was simulated with initial conditions (x0,y0,z0)=(0,0,0), (φ0,θ0,Ψ0)=(0,0,0), and (θ10,θ20)=(0,0). The desired position, attitude, and joint angles were set as (xd,yd,zd)=(30,50,40)[m], (φd,θd,Ψd)=(0,0,0), and (θ1d,θ2d)=(30,60)[°], respectively.

The control parameters are chosen as

(k1,k2,k3,k4,k5,k6)=(2,0.1,2,0.1,2,2),E80
(λ1,λ2)=(1,1),(α1,α2)=(0.1,0.1).E81

As shown in Figures 35, the position, attitude, and joint angles converge to their desired values in around 40 s. Figure 6 shows the time responses of the control inputs ui, i = 1,…,4.

Figure 3.

Time responses of the aerial robot’s position x, y, and z.

Figure 4.

Time responses of the aerial robot’s Euler angles φ, θ, and Ψ.

Figure 5.

Time responses of the robotic arm’s joint angles θ1 and θ2.

Figure 6.

Time responses of the control inputs ui, i = 1,…,4.

5. Conclusions

This chapter first derives the complete nonlinear dynamics of an aerial robot consisting of a quadcopter with a two-link robot manipulator. Precise control of such an aerial robot is a challenging task since the translational and rotational dynamics of the quadcopter are strongly coupled with the dynamics of the manipulator. We extend our previous results on the control of quadrotor UAVs to the control of aerial robots. In particular, we design a backstepping and Lyapunov-based nonlinear feedback control law that achieves the point-to-point control of the areal robot. The effectiveness of this feedback control law is illustrated through a simulation example.

The many avenues considered for future research include problems involving collaborative control of multiple aerial robots. Future research also includes designing nonlinear control laws that achieve robustness, insensitivity to system and control parameters, and improved disturbance rejection. We also plan to explore the use of geometric mechanics formulation of such control problems.

The matrices M and J¯can be expressed as

M=mt[100010001],J¯=[J¯11J¯12J¯13J¯21J¯22J¯23J¯31J¯32J¯33],

where

J¯11=Jxx+[m1lc12+(m2+mp)l12+112m1l12]sin2θ1+[m2lc22+mpl22+112m2l22]sin2θ2+2l1(m2lc2+mpl2)sinθ1sinθ2,J¯22=Jyy+m1lc12+(m2+mp)l12+(m2lc22+mpl22)+2l1(m2lc2+mpl2)cos(θ2θ1)+112[m1l12+m2l22],J¯33=Jzz+[m1lc12+(m2+mp)l12+112m1l12]cos2θ1+[m2lc22+mpl22+112m2l22]cos2θ2+2l1(m2lc2+mpl2)cosθ1cosθ2,J¯12=J¯21=J¯23=J¯32=0,J¯13=J¯31=12[m1lc12+(m2+mp)l1+112m1l12]sin2θ1l1(m2lc2+mpl2)sin(θ1+θ2)12[(m2lc22+mpl22)+112m2l22]sin2θ2.

The matrix m¯can be computed as

m¯=[m1lc12+(m2+mp)l12+112m1l12(m2lc2+mpl2)l1cos(θ2θ1)(m2lc2+mpl2)l1cos(θ2θ1)m2lc22+mpl22+112m2l22].

The matrices K, Br, and Bt are given by

K=[0Kxy0Kxy0Kxz0Kxz0],Br=[00Br1Br200],

where

Kxy=[m1lc1+(m2+mp)l1]sinθ1+(m2lc2+mpl2)sinθ2,Kxz=[m1lc1+(m2+mp)l1]cosθ1+(m2lc2+mpl2)cosθ2,Br1=[m1lc12+(m2+mp)l12+(m2lc2+mpl2)l1cos(θ2θ1)]+112m1l12,Br2=[m2lc22+mpl22+(m2lc2+mpl2)l1cos(θ2θ1)]+112m2l22,

and

Bt=[(m1lc1+m2l1+mpl1)sinθ1(m2lc2+mpl2)sinθ200(m1lc1+m2l1+mpl1)cosθ1(m2lc2+mpl2)cosθ2].

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Mahmut Reyhanoglu and Muhammad Rehan (September 6th 2017). Nonlinear Dynamics and Control of Aerial Robots, Aerial Robots - Aerodynamics, Control and Applications, Omar Dario Lopez Mejia and Jaime Alberto Escobar Gomez, IntechOpen, DOI: 10.5772/intechopen.69641. Available from:

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