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# Nonlinear Dynamics and Control of Aerial Robots

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Reviewed: May 8th, 2017 Published: September 6th, 2017

DOI: 10.5772/intechopen.69641

From the Edited Volume

## Aerial Robots

Edited by Omar Dario Lopez Mejia and Jaime Alberto Escobar Gomez

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## 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

• 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].

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 v R 3 , ω R 3 , and η S 1 × S 1 denote 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:

d d t T ν + ω ^ T ν = F t , E1
d d t T ω + ω ^ T ω + ν ^ T ν = τ r , E2
d d t T η ˙ T η = τ s , E3

where Ft R 3 , τr R 3 denote the vectors of generalized control forces and generalized control torques, respectively, that act on the base body, and τs R 3 is the vector of joint torques. For a given vector a = [a1 a2 a3]T R 3 , the skew-symmetric matrix a ^ defines the corresponding cross-product operation a×, given by

a ^ = [ 0 a 3 a 2 a 3 0 a 1 a 2 a 1 0 ] . 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 F 1 and by xB yB zB axes the body-fixed frame F B 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 F B. Then, the translational and rotational kinematics can be expressed as

p ˙ = R v , E5
R ˙ = R ω ^ . 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

F i = b Ω i 2 = b K ν 2 V i 2 , 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

F p = i = 1 4 F i e 3 = b K v 2 ( V 1 2 + V 2 2 + V 3 2 + V 4 2 ) e 3 , E8

where e3 = [0 0 1]T R 3 is 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.

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 l c i denote 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

ρ 1 = [ l c 1 cos θ 1 0 l c 1 sin θ 1 ] T , E9
ρ 2 = [ l 1 cos θ 1 + l c 2 cos θ 2 0 l 1 sin θ 1 + l c 2 sin θ 2 ] T , E10
ρ p = [ l 1 cos θ 1 + l 2 cos θ 2 0 l 1 sin θ 1 + l 2 sin θ 2 ] 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

F g = m t g R T e 3 , E12
τ g = g [ m 1 ρ 1 + m 2 ρ 2 + m p ρ p ] × R T e 3 , 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 ( F 4 F 2 ) = b l K ν 2 ( V 4 2 V 2 2 ) about the roll axis.

• Propellers 1 and 3 generate a moment l ( F 3 F 1 ) = b l K ν 2 ( V 3 2 V 1 2 ) about the pitch axis.

• The sum of all torques about z-axis is d K ν 2 ( V 1 2 V 2 2 + V 3 2 V 4 2 ) and causes a yaw moment.

• The rotation of the propellers causes the gyroscopic effect J r K ν ω y ( V 1 V 2 + V 3 V 4 ) about the roll-axis and J r K ν ω x ( V 1 V 2 + V 3 V 4 ) 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

τ p = [ b l K ν 2 ( V 4 2 V 2 2 ) + J r ω y Ω r b l K ν 2 ( V 3 2 V 1 2 ) J r ω x Ω r d K ν 2 ( V 1 2 V 2 2 + V 3 2 V 4 2 ) ] , E14

where Ω r : = K ν ( V 1 V 2 + V 3 V 4 ) is the overall residual angular speed of the propellers.

Let η = [ θ 1 θ 2 ] T denote the shape variables. Then, the linear and angular velocities of each link, and the linear velocity of the payload can be expressed in F B as

ν i = ν + ω ^ ρ i + ρ i η η ˙ = ν ρ ^ i ω + ρ i η η ˙ , i = 1,2, E15
ω i = ω + θ ˙ i e 2 = ω + C i ( η ) η ˙ , i = 1,2, E16
ν p = ν + ω ^ ρ p + ρ p η η ˙ = ν ρ ^ p ω + ρ p η η ˙ , E17

where e2 = [0 1 0]T R 3 is the second standard basis vector. The total kinetic energy can now be expressed as

T ( ν , ω , η , η ˙ ) = 1 2 m ν T ν + 1 2 ω T J ω + 1 2 i = 1 2 ( m i ν i T ν i + ω i T J i ω i ) + 1 2 m p ν p 2 , E18

where J i = R i T J ¯ i R i is 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

R i = [ cos θ i 0 sin θ i 0 1 0 sin θ i 0 cos θ i ] , i = 1,2, E19

and J ¯ i denotes 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 ¯ i can be expressed as

J ¯ i = [ 0 0 0 0 1 12 m i l i 2 0 0 0 1 12 m i l i 2 ] . E20

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

[ M K B t K T J ¯ B r B t T B r T m ¯ ] [ v ˙ w ˙ η ¨ ] = [ F t τ r τ s ] [ w ^ M K ˙ + w ^ K B ˙ t + w ^ B t K ˙ T + w ^ K T + ν ^ M J ¯ ˙ + w ^ J ¯ + ν ^ K B ˙ a + w ^ B a + ν ^ B t B ˙ t T B ˙ a T m ¯ ˙ ] [ v w η ˙ ] + [ 0 0 L η ] , E21

where

M = m t I 3 × 3 , E22
J ¯ = J + m 1 ρ ^ 1 T ρ ^ 1 + m 2 ρ ^ 2 T ρ ^ 2 + m p ρ ^ p T ρ ^ p + J 1 + J 2 , E23
m ¯ = m 1 ρ 1 η T ρ 1 η + m 2 ρ 2 η T ρ 2 η + m p ρ p η T ρ p η + C 1 T J 1 C 1 + C 2 T J 2 C 2 , E24
K = m 1 ρ ^ 1 m 2 ρ ^ 2 m p ρ ^ p , E25
B t = m 1 ρ 1 η + m 2 ρ 2 η + m p ρ p η , E26
B r = m 1 ρ ^ 1 ρ 1 η + m 2 ρ ^ 2 ρ 2 η + m p ρ ^ p ρ p η + J 1 C 1 + J 2 C 2 . 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

M ν ˙ + K ω ˙ + B t η ¨ = F t ω ^ M ν ( K ˙ + ω ^ K ) ω ( B ˙ t + ω ^ B t ) η ˙ , E28
K T ν ˙ + J ¯ ω ˙ + B a η ¨ = τ r ( K ˙ T + ω ^ K T + ν ^ M ) ν ( J ¯ ˙ + ω ^ J ¯ + ν ^ K ) ω ( B ˙ a + ω ^ B a + ν ^ B t ) η ˙ , E29
B t T ν ˙ + B a T ω ˙ + m ¯ η ¨ = τ s B ˙ t T ν B ˙ a T ω m ¯ ˙ η ˙ + L η . E30

Eq. (28) can be rewritten as

M ν ˙ = F t ω ^ M ν + F d , E31

where

F d = K ω ˙ B t η ¨ ( K ˙ + ω ^ K ) ω ( B ˙ t + w ^ B t ) η ˙ . E32

Eq. (31) can be simplified as

[ x ¨ y ¨ z ¨ ] = [ 0 0 g ] + R e ^ 3 u 1 + F ¯ d . E33

where F ¯ d = F d / m t and

u 1 = b K v 2 ( V 1 2 + V 2 2 + V 3 2 + V 4 2 ) / m t , E34

Similarly, Eq. (29) can be rewritten as

J ¯ ω ˙ = τ r ( J ¯ ˙ + ω ^ J ¯ ) ω + τ d , E35

where

τ d = K T v ˙ B a η ¨ ( K ˙ T + ω ^ K T + ν ^ M ) ν ν ^ K ω ( B ˙ a + ω ^ B a + ν ^ B t ) η ˙ . E36

Eq. (35) can be simplified as

ω ˙ = J ¯ 1 τ r J ¯ 1 ( J ¯ ˙ + ω ^ J ¯ ) ω + τ ¯ d , E37

where τ ¯ d = J ¯ 1 τ d .

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

x ¨ = ( cos φ sin θ cos Ψ + sin φ sin Ψ ) u 1 , E38
y ¨ = ( cos φ sin θ sin Ψ sin φ cos Ψ ) u 1 , E39
z ¨ = g + ( cos φ cos θ ) u 1 , E40
[ ω ˙ x ω ˙ y ω ˙ z ] = J ¯ 1 ( τ g + [ J r ω y Ω r J r ω x Ω r 0 ] ( J ¯ ˙ + ω ^ J ¯ ) ω ) + [ u 2 u 3 u 4 ] , E41

where

[ u 2 u 3 u 4 ] = J ¯ 1 [ b l K ν 2 ( V 4 2 V 2 2 ) b l K ν 2 ( V 3 2 V 1 2 ) b l K ν 2 ( V 1 2 V 2 2 + V 3 2 V 4 2 ) ] . 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 ˙ ] T and ω ˙ [ φ ¨ θ ¨ Ψ ¨ ] T , and hence the equation of motions can be simplified as

x ¨ = θ u 1 , E43
y ¨ = φ u 1 , E44
z ¨ = g + u 1 , E45
φ ¨ = f 1 ( φ , θ , Ψ ) + u 2 , E46
θ ¨ = f 2 ( φ , θ , Ψ ) + u 3 , E47
Ψ ¨ = f 3 ( φ , θ , Ψ ) + u 4 , E48

where

[ f 1 f 2 f 3 ] = J ¯ 1 ( τ g + [ J r θ ˙ Ω r J r φ ˙ Ω r 0 ] ( 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 ) = ( x d , y d , z d , φ d , θ d , Ψ d ) .

We choose u1 as

u 1 = g | z z d | a s i g n ( z z d ) | z ˙ | b s i g n ( 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 ˙ ) = ( z d , 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 ¨ = g θ , E51
y ¨ = g φ . 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 ) = ( x d , y d , φ d , θ d ) .

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 = k 1 ( x x d ) k 2 x ˙ , E53
φ = k 3 ( y y d ) + k 4 y ˙ , E54

where ki > 0, i = 1,…,4, so that

x ¨ + g k 2 x ˙ + g k 1 ( x x d ) = 0, E55
y ¨ + g k 3 y ˙ + g k 4 ( y y d ) = 0. E56

Define

y 1 = θ + k 1 ( x x d ) + k 2 x ˙ , E57
y 2 = φ k 3 ( y y d ) k 4 y ˙ , E58

and consider the y1 and y2 dynamics given by

y ˙ 1 = θ ˙ + k 1 x ˙ + k 2 x ¨ = θ ˙ + k 1 x ˙ + k 2 g θ , E59
y ˙ 2 = φ ˙ k 3 y ˙ k 4 y ¨ = φ ˙ k 3 y ˙ + k 4 g φ . E60

Define the sliding variables (s1, s2)

s 1 = y ˙ 1 + α 1 y 1 , E61
s 2 = y ˙ 2 + α 2 y 2 , E62

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

s 1 = θ ˙ + ( k 2 g + α 1 ) θ + ( k 1 + α 1 k 2 ) x ˙ + α 1 k 1 ( x x d ) , E63
s 2 = φ ˙ + ( k 4 g + α 2 ) φ ( k 3 + α 2 k 4 ) y ˙ α 2 k 3 ( y y d ) . E64

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

s ˙ 1 = θ ¨ + ( k 2 g + α 1 ) θ ˙ + ( k 1 + α 1 k 2 ) g θ + α 1 k 1 x ˙ , E65
s ˙ 2 = φ ¨ + ( k 4 g + α 2 ) φ ˙ + ( k 3 + α 2 k 4 ) g φ α 2 k 3 y ˙ . E66

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

s ˙ 1 = f 2 ( φ , θ , Ψ ) + u 3 + ( k 2 g + α 1 ) θ ˙ + ( k 1 + α 1 k 2 ) g θ + α 1 k 1 x ˙ , E67
s ˙ 2 = f 1 ( φ , θ , Ψ ) + u 2 + ( k 4 g + α 2 ) φ ˙ + ( k 3 + α 2 k 4 ) g φ α 2 k 3 y ˙ . E68

We choose the inputs u2 and u3 as

u 2 = λ 2 sign ( s 2 ) f 1 ( φ , θ , Ψ ) ( k 4 g + α 2 ) φ ˙ ( k 3 + α 2 k 4 ) g φ + α 2 k 3 y ˙ , E69
u 3 = λ 1 sign ( s 1 ) f 2 ( φ , θ , Ψ ) ( k 2 g + α 1 ) θ ˙ ( k 1 + α 1 k 2 ) g θ α 1 k 1 x ˙ , E70

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

s ˙ 1 = λ 1 sign ( s 1 ) , E71
s ˙ 2 = λ 2 sign ( s 2 ) , 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 ˙ ) = ( Ψ d , 0 ) :

u 4 = k 5 ( Ψ Ψ d ) k 6 ψ ˙ f 3 ( 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 ˙ = M 1 F t M 1 w ^ M ν , E74
ω ˙ = J ¯ 1 τ r J ¯ 1 ( J ¯ ˙ + ω ^ J ¯ ) ω . E75

Eq. (30) can be rewritten as

η ¨ = m ¯ 1 ( B t T ν ˙ B a T w ˙ + τ s B ˙ t T ν B ˙ a T w m ¯ ˙ η ˙ + L η ) , E76

which can be expressed in terms of Ft and τr as

η ¨ = m ¯ 1 ( τ s B t T F t m t B t T ω ^ ν B a T J ¯ 1 ( τ r ( J ¯ ˙ + ω ^ J ¯ ) ω ) B ˙ t T ν B ˙ a T w m ¯ ˙ η ˙ + L η ) . E77

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

τ s = B t T F t m t + B t T ω ^ v + B a T J ¯ 1 ( τ r ( J ¯ ˙ + ω ^ J ¯ ) ω ) + B ˙ t T ν + B ˙ a T w + m ¯ ˙ η ˙ L η m ¯ ( 2 λ η ˙ + λ 2 ( η η d ) ) E78

where λ > 0, so that

η ¨ + 2 λ η ˙ + λ 2 ( η η d ) = 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.

Symbol Parameter Value Unit
Kv Transformation constant 54.945 rad s V−1
Jr Rotor inertia 6 × 10−5 kg m2
Jxx MOI about x axis 0.0552 kg m2
Jyy MOI about y axis 0.0552 kg m2
Jzz MOI about z axis 0.1104 kg m2
b Thrust coefficient 3.935139 × 10−6 N V−1
d Drag coefficient 1.192564 × 10−7 Nm V−1
l Distance from pivot to motor 0.1969 m
m Mass 2.85 kg
g Acceleration of gravity 9.81 ms−2
V Maximum input voltage 10 V
m1 Mass of link 1 0.1 kg
m2 Mass of link 2 0.1 kg
mp Mass of the payload 0.1 kg
l1 Length of link 1 0.5 m
l2 Length of link 2 0.5 m

### Table 1.

Parameters of the aerial robot.

A rest-to-rest motion was simulated with initial conditions ( x 0 , y 0 , z 0 ) = ( 0, 0, 0 ) , ( φ 0 , θ 0 , Ψ 0 ) = ( 0, 0, 0 ) , and ( θ 10 , θ 20 ) = ( 0, 0 ) . The desired position, attitude, and joint angles were set as ( x d , y d , z d ) = ( 30, 50, 40 ) [m], ( φ d , θ d , Ψ d ) = ( 0, 0, 0 ) , and ( θ 1 d , θ 2 d ) = ( 30, 60 ) [°], respectively.

The control parameters are chosen as

( k 1 , k 2 , k 3 , k 4 , k 5 , k 6 ) = ( 2, 0.1, 2, 0.1, 2, 2 ) , E80
( λ 1 , λ 2 ) = ( 1, 1 ) , ( α 1 , α 2 ) = ( 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.

## 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 = m t [ 1 0 0 0 1 0 0 0 1 ] , J ¯ = [ J ¯ 11 J ¯ 12 J ¯ 13 J ¯ 21 J ¯ 22 J ¯ 23 J ¯ 31 J ¯ 32 J ¯ 33 ] ,

where

J ¯ 11 = J x x + [ m 1 l c 1 2 + ( m 2 + m p ) l 1 2 + 1 12 m 1 l 1 2 ] sin 2 θ 1 + [ m 2 l c 2 2 + m p l 2 2 + 1 12 m 2 l 2 2 ] sin 2 θ 2 + 2 l 1 ( m 2 l c 2 + m p l 2 ) sin θ 1 sin θ 2 , J ¯ 22 = J y y + m 1 l c 1 2 + ( m 2 + m p ) l 1 2 + ( m 2 l c 2 2 + m p l 2 2 ) + 2 l 1 ( m 2 l c 2 + m p l 2 ) cos ( θ 2 θ 1 ) + 1 12 [ m 1 l 1 2 + m 2 l 2 2 ] , J ¯ 33 = J z z + [ m 1 l c 1 2 + ( m 2 + m p ) l 1 2 + 1 12 m 1 l 1 2 ] cos 2 θ 1 + [ m 2 l c 2 2 + m p l 2 2 + 1 12 m 2 l 2 2 ] cos 2 θ 2 + 2 l 1 ( m 2 l c 2 + m p l 2 ) cos θ 1 cos θ 2 , J ¯ 12 = J ¯ 21 = J ¯ 23 = J ¯ 32 = 0, J ¯ 13 = J ¯ 31 = 1 2 [ m 1 l c 1 2 + ( m 2 + m p ) l 1 + 1 12 m 1 l 1 2 ] sin 2 θ 1 l 1 ( m 2 l c 2 + m p l 2 ) sin ( θ 1 + θ 2 ) 1 2 [ ( m 2 l c 2 2 + m p l 2 2 ) + 1 12 m 2 l 2 2 ] sin 2 θ 2 .

The matrix m ¯ can be computed as

m ¯ = [ m 1 l c 1 2 + ( m 2 + m p ) l 1 2 + 1 12 m 1 l 1 2 ( m 2 l c 2 + m p l 2 ) l 1 cos ( θ 2 θ 1 ) ( m 2 l c 2 + m p l 2 ) l 1 cos ( θ 2 θ 1 ) m 2 l c 2 2 + m p l 2 2 + 1 12 m 2 l 2 2 ] .

The matrices K, Br, and Bt are given by

K = [ 0 K x y 0 K x y 0 K x z 0 K x z 0 ] , B r = [ 0 0 B r 1 B r 2 0 0 ] ,

where

K x y = [ m 1 l c 1 + ( m 2 + m p ) l 1 ] sin θ 1 + ( m 2 l c 2 + m p l 2 ) sin θ 2 , K x z = [ m 1 l c 1 + ( m 2 + m p ) l 1 ] cos θ 1 + ( m 2 l c 2 + m p l 2 ) cos θ 2 , B r 1 = [ m 1 l c 1 2 + ( m 2 + m p ) l 1 2 + ( m 2 l c 2 + m p l 2 ) l 1 cos ( θ 2 θ 1 ) ] + 1 12 m 1 l 1 2 , B r 2 = [ m 2 l c 2 2 + m p l 2 2 + ( m 2 l c 2 + m p l 2 ) l 1 cos ( θ 2 θ 1 ) ] + 1 12 m 2 l 2 2 ,

and

B t = [ ( m 1 l c 1 + m 2 l 1 + m p l 1 ) sin θ 1 ( m 2 l c 2 + m p l 2 ) sin θ 2 0 0 ( m 1 l c 1 + m 2 l 1 + m p l 1 ) cos θ 1 ( m 2 l c 2 + m p l 2 ) cos θ 2 ] .

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