PD Sliding Mode Controller based Decoupled Aerial Manipulation

Kamel Bouzgou

1,2 a

, Laredj Benchikh

1 b

, Lydie Nouveliere

1 c

, Yasmina Bestaoui

1 d

and Zoubir Ahmed-Foitih

2 e

IBISC, Univ. Evry, Universit

e Paris-Saclay, 91025, Evry, France

LEPESA Laboratory, Faculty of Electrical Engineering, Department of Electronics, USTO-MB,31000 Oran, Algeria

Keywords:

Aerial Manipulation, Decoupled Dynamic System, SMC, Sliding Mode Control, PD-SMC.

Abstract:

This paper presents the design of 3-Dof multi-link robot arm that is mounted on the multirotor. To be consid-

ered the dynamic characteristics of the manipulation platform, the decoupled dynamic models of the system

are derived. The main advantage of the ﬁrst joint is introduced for more robustness and stability during hover-

ing. The PID controller will be implemented for position and attitude of multirotor control, whereas, a sliding

mode controller will be designed for a manipulator robot, which is then compared with the sliding surface

that has been integrated with the proportional-derivative (PD) controller. The performance of the proposed

technique is demonstrated through a simulation using Simulink and Matlab environment.

1 INTRODUCTION

Unmanned aerial manipulators (UAMs) are used for

mobile manipulations thanks to their mobility which

enables a simple hovering to aggressive manoeuvr-

ers. It has emerged a need for the interaction of that

UAVs with the environment that is not easily acces-

sible by humans, for this aim, the researchers have

used this structure for transporting, manipulation and

grasping tasks. Moreover, the control of these aerial

manipulator is a very challenging problem, consider-

ing that such system is under-actuation and the cou-

pled dynamics between the two different platforms,

and the contact with the environment during manipu-

lation. Several survey papers deal with projects that

the multi-link robotic arm attached to the UAV for

manipulation tasks are used, the dynamic formalism

and control technique developed. Authors in (Bouz-

gou et al., 2019a) have classiﬁed aerial manipulator

systems based on aerial vehicles and the attached arm

kind. For the simple grasping task, a magnet attached

to the UAV is presented in (Escareno et al., 2014). A

single Dof to push an object in the desired direction,

in (Srikanth et al., 2011; Yeol et al., 2017) is designed

https://orcid.org/0000-0003-2374-2149

https://orcid.org/0000-0002-4617-399X

https://orcid.org/0000-0003-0027-7192

https://orcid.org/0000-0001-7716-5952

https://orcid.org/0000-0003-3121-9964

Figure 1: The structure of Q-PRR with principal frames.

as a non-prehensile manipulation and tentacle system.

The most structure of aerial manipulator arm is

when a 2-Dof robot arm iw mounted (Aydemir et al.,

2015; Kim et al., 2013), that number of dof can help

users to avoid a singularity region in the manipulator

arm workspace. A 3-Dof manipulator arm as revo-

lute joints is used in (Mello et al., 2015), when 4-Dof

multi-link manipulator arm is presented in (Jimenez-

Cano et al., 2017) as a serial robot placed at the up-

per part of the UAV for bridge inspection. In (Kon-

dak et al., 2013) authors are used a 5-Dof manipulator

robot for environment interaction. An industrial ma-

nipulator as a redundant robot arm with n > 6 is used

in (Huber et al., 2013) aerial manipulation with a 7-

Dof based on a main-tail-rotor helicopter, in (Danko

484

Bouzgou, K., Benchikh, L., Nouveliere, L., Bestaoui, Y. and Ahmed-Foitih, Z.

PD Sliding Mode Controller based Decoupled Aerial Manipulation.

DOI: 10.5220/0009856704840489

In Proceedings of the 17th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2020), pages 484-489

ISBN: 978-989-758-442-8

and Oh, 2014) the Hyper-redundant manipulator is

presented as 9-Dof for a wide reachable workspace.

A delta structure ﬁxed on side of UAV was introduced

in (Fumagalli et al., 2014), when a parallel robot is

used in (Cho and Shim, 2017; Danko et al., 2015).

For an object interaction and manipulation applying

forces and torques, a dual 4-dof arm mounted on UAV

is presented in (Korpela et al., 2013).

Researchers in (Ruggiero et al., 2015), have de-

veloped a structure with a moving battery of the UAV

in one direction in order to hold the Multirotor CoG

as close as possible to the vertical axis, The drawback

of such structure is that battery movement is bounded

when the end-effector tried to reach the desired po-

sition and battery position cannot always ensure the

alignment of CoM of UAV and robot arm. A new

aerial manipulator with a 3-Dof is developed in this

paper (Bouzgou et al., 2019b), the alignment of CoG

of whole system can be ensured with a simple move-

ment of a prismatic joint along x-axis Figure 1. wide

workspace with this design and offering large possible

conﬁgurations of the robot arm for a desired position

and attitude without losing the optimum location of

the Q-PRR CoG (Bouzgou and Ahmed-Foitih, 2015).

An adaptive sliding controller based on a tradi-

tional Lagrange modeling method was proposed(Kim

et al., 2013), An augmented adaptive sliding con-

troller based on a closed-chain robot dynamics was

presented for cooperative transportation of multiple

(Lee et al., 2015) online estimation of objects based

on an augmented adaptive sliding controller was pro-

posed.

2 MODELING

The conﬁguration of the proposed aerial manipulator

consists of two parts, the multi-rotor with the num-

ber of rotors that n

= 4 that can be called with the

letter ”Q” for Quadrotor more details see (Bouzgou

et al., 2019a), and the manipulator arm attached to the

bottom, whose geometric centers are considered to be

in the same z axis (

) of the mobile frame

B. The

manipulator consists of three degrees of freedom (3-

Dof), Prismatic-Revolute-Revolute joints, called (Q-

PRR), the ﬁrst joint is prismatic and its axis (x

) is

parallel to the x-axis of the multirotor mobile frame

B, this joint is considered actuated and it moves along

the same axis, and it is bounded on both directions

by a value r

, the distance between two axes x

and

is denoted by (d

), The second and third joints are

revolute, their rotation axes (z

)and (z

)are parallel to

the (y) axis of the

B unit, where the manipulator arm

movements are considered in the (x, z)-axis of the

mobile frame. The position of the

B ﬁxed frame with

respect to the

E inertial frame is given by the (3 × 1)

vector denoted by p

, whereas its orientation denoted

by R

is represented by the XY Z rotation sequence

with the ﬁxed frame axes, and its orientation by the

ﬁxed frame axes.





− c

+ s

+ c

− s

−s





(1)

where s

∗

= sin(∗), c

∗

= cos(∗), Let R

be the orien-

tation matrix of the end-effector attached frame, and





the position vector of the origin

with respect to

B ﬁxed frame, and the absolute posi-

tion vector and orientation matrix of the end-effector

with respect to

E, is given by the p





and

, respectively, where the pair (p

) ∈ SE(3) de-

notes the vector position given by p





and the orientation matrix of the multirotor with re-

spect to the inertial frame

The position and orientation equations of end-

effector expressed in

E are written as follows:

= p

+ R

(2)

= R

(3)

By denoting with

the time derivative of ϕ

= T (φ

)

(4)

Where T (φ

) is the transformation matrix be-

tween the time derivative of the Euler angles ϕ

and

the angular velocity of the multirotor ω

T (φ

) =





1 0 −s

0 c

0 −s





(5)

By differentiating (2), (3) and with taking into ac-

count (4), the translational and angular velocities of

the end-effector with respect to

E can be written as

follows:

˙p

= ˙p

− R

T (φ

)

+ R

˙p

(6)

= T (φ

)

+ R

(7)

Where ˙p

,ω

are the linear and angular velocities

of the mobile frame

B with respect to the

E frame,

respectively, and ˙p

,ω

are the translational and an-

gular velocities of the end-effector with respect to the

mobile frame

B. (ˆ.), the hat map that transforms a

vector in R

to (3 × 3) Skew-symmetric matrix such

that ˆxy = x × y, ∀x, y ∈ R

(Kamel et al., 2017).

The dynamic model of Q-PRR can be derived by

considering the Lagrange formulation in details in

(Lippiello and Ruggiero, 2012). The function of La-

grangian is then expressed by L = E − U where E ,

PD Sliding Mode Controller based Decoupled Aerial Manipulation

485

U denote the kinematics and potential energy of the

whole system, respectively. The Lagrange equations

are given by

δL

−

δL

δξ

= u + u

ext

(8)

Where i = 1, ..., 6 + n is the i − th coordinate of ξ,

and u

is the ((6 +n)×1) vector of generalized forces

and torques, and u

ext

denotes the vector of external

disturbance forces and torques. the dynamic model of

the global system can be written as

B(ξ)

ξ +C(ξ,

ξ)

ξ + G(ξ) = u + u

ext

(9)

Where G is a ((6 + n) × 1) vector of gravitational

terms given by deriving the potential energy as

G(ξ) =

δP

δξ

And C is the matrix of Coriolis and cen-

trifugal terms. more details see (Bouzgou et al.,

2019b).

is the vector of generalized forces at the i-th

joint level. τ

is the vector of torques composed with

, τ

and τ

generated by the system computed

around x,y and z axes respectively. Let µ be the vector

of the force f

applied to the prismatic joint and τ

torques applied by the revolute joint of the robot

arm.

u =

















=diag(R

,Q, I

)









, τ









, µ =



















1 1 1 1

0 l 0 −l

−l 0 l 0

c −c c −c



















(10)

Where l is the distance from each motor to the mul-

tirotor centre of mass. When c is the drag factor.

Since B(ξ

) is symmetric B

i j

= B

, and Let the ma-

trices M

, M

pϕ

, M pq, M

ϕϕ

, M

ϕq

, M

be deﬁned

by partitioning the mass matrix and by according to

the equation deﬁned in Equation 9 the decoupled dy-

namic model for the overall system is presented and a

sub-matrices are given from that equation and can be

rewrite as follows

B(ξ

) =





pϕ

ϕϕ

ϕq





(11)

C(ξ,

ξ) =









, G(ξ

) =









The dynamic modeling of the decoupled system is

presented as follows For the multirotor, a new coordi-

nate joint can be deﬁned as a part of the generalized

coordinates xi

such that q





. Multirotor

motion equations Equations to respect the Lagrange

formalism in Equation 9 and the decomposition of the

B,C and G matrices can be deﬁned as follows

¨p

+ B

pϕ

˙p

+ G

+ B

¨q

= u

(12)

pϕ

¨p

+ B

ϕϕ

+ G

+ B

ϕq

¨q

= u

(13)

¨p

+ B

ϕq

˙q

+ G

+ B

¨q

= u

(14)

From the equation 12 and 13, the dynamic model

can be simpliﬁed with taking into account the de-

coupled terms. Rewrite that with the model deﬁned

in (Kamel et al., 2017), where M

= B

¨q

, M

ϕq

¨q

and M

= B

¨p

+ B

ϕq

∈ R

3×1

, such as





and M





Equations becomes as

m ¨x = (c

+ s

− M

(15)

m ¨y = (c

− s

− M

(16)

m¨z = (c

− mg − M

(17)

φ =

ψ(I

− I

) − J

θΩ

+ lu

− M

(18)

θ =

ψ(I

− I

) − J

ψΩ

+ lu

− M

(19)

ψ =

θ(I

− I

) + u

− M

(20)

where, Ω

= (ω

+ ω

− ω

), I

, I

are

the inertia matrix terms and m the multirotor mass.

The dynamic modeling of the robot arm is deﬁned

using a Lagrange Formalism depicted in Equation

9, where the jacobians matrix are deﬁned by using

a Denavit-Hartenberg method depicted in (Bouzgou

and Ahmed-Foitih, 2014)and its equation of move-

ment can be presented as follow

= (m

+ m

)¨r



(

+θ

+ d

) + d



+θ

)

−



(

+θ

+ d

) + d



−

+θ

− d

+θ

− g(m

+ m

) (21)

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

486

= (m

(

+θ

+ d

) +

)¨r



+ I

+ d





(θ

)

+ I



−

2θ

+2θ

− I

2θ

+2θ

+ d

)

− d

− g



(

+θ

+ d

) +



(22)

+θ

¨r







+ I

+θ

− I

+θ



(θ



+θ

− 2I

+θ



+ s

2θ

+2θ

− I

)

−

+θ

(23)

Where, c

∗

= cos(∗),s

∗

= sin(∗). These equations can

be rewritten as a matrix forms regarding to the Equa-

tion 14.

3 CONTROL DESIGN

This section describes the design of the controller for

the decoupled system, it is composed into three parts,

attitude control of the multirotor as a function of the

position of the desired center of gravity; and con-

trol of the manipulator arm when the system reaches

the target position point, the PID proportional integral

derivative controller is used for the multirotor and the

sliding mode technique for the manipulator arm and

then it will be improve when the PID will be added

in parallel. Figure 2 shows the diagram of the control

strategy.

From the dynamic equations , we have a de-

coupled system where the translational accelerations

don’t depend on angular acceleration, from the ﬁrst

three equations we can extract the pitch and roll an-

gles when we introduce the desired position and atti-

tude. From equations 15, 16, 17, θ and φ can be com-

puted and its depended on multirotor position vector

Figure 2: Diagram of the control strategy.

and altitude



x y z ψ



, therefore, it can be writing as

follows

tan(θ) =



¨x +





¨y +



¨z + g +

(24)

sin(φ) =



¨x +





¨y +





¨x +





¨y +





¨z + g +



(25)

When the reference value for the roll and pitch an-

gles has been calculated, the control inputs of those

angles can be determined, and the feedback of the

speed time derivative will be required to determine

the input vector, in the continuous time,the parallel

PID controller is deﬁned, it have a following form

u(t) = K

pζ

e(t) + K

iζ

e(τ)dτ + K

dζ

de(t)

(26)

ζ is the variable to be controlled. ζ ∈ {x, y, z, φ, θ, ψ}

Where,e(t) = ζ

− ζ, is the error between the desired

and measured value of ζ and K

pζ

, K

iζ

are Proportional

and Integral gain,respectively.

3.1 Manipulator Arm Controller

For the manipulator arm, sliding mode control strat-

egy will be presented, the main idea of this approach

can be presented as follows.

Considering the non-linear n-order object deﬁned

(n)

(t) = f (x,t) + u(t) (27)

where x is the state variable,u is the control law. Let

be e is the error between desired and measured value

of x, such as e = x

des

− x, let be s = 0 as a sliding

surface deﬁned by

s = e

(n−1)

+ λ

(n−2)

+ ... + λ

n−2

˙e + λ

n−1

.e (28)

Where λ is the coefﬁcient of the sliding surface.

Derivative the equation 28, it can write

˙s = e

+ λ

(n−1)

+ ... + λ

n−2

+ λ

n−1

. ˙e (29)

Using Equation 27 and with choosing the input con-

trol vector u that ˙s = Ksgn(s) with K is positive de-

ﬁned. Substituting into Equation 29, the following

PD Sliding Mode Controller based Decoupled Aerial Manipulation

487

equation can be rewritten

u = − f (x,t)+x

(n)

des

+λ

(n−1)

+...+λ

n−1

. ˙e+Ksgn(s)

(30)

sgn is the sign function, applying equation 30 with

n = 2 and for three link robot arm ¨x = f (x,t) + u(t).

The input control law is as follows

u(t) = − f (x,t) + q

des

+ λ. ˙e + Ksgn(s) (31)

With q

des



0,des

2,des

3,des



, λ =





and K =





4 SIMULATION AND RESULTS

Simulations were conducted to evaluate the perfor-

mance of the aerial manipulator in maintaining an

end-effector position relative to ﬁxed targets. the de-

sired position of the multirotor is given by p

[m] =



20 30 10



, and when the roll and pitch angles of the

multirotor are considering as zero, the movement of

the arm will be started, and the desired trajectory of all

joints is given by p



0.04 0.6 0.2



[m,rad,rad]

The simulation results are summarized in Figure 3.

(a) (b)

Figure 3: Position of the multirotor centre of gravity in Fig-

ure 3a, the orientation of the multirotor ﬁxed frame

B to

respect to the reference frame

E in the Figure 3b.

The position errors of the end-effector can be sum-

marized in the Figure 4.

Figure 4: Position Errors of the CoM of Q-PRR.

Figure 5: Position of the Center of mass of the Q-PRR.

Figure 6: Response of manipulator joints.

5 CONCLUSIONS

In this paper, a mathematical model for a decoupled

system (multirotor and robot arm) is presented by us-

ing the Denavit-Hartenberg approach for geometric

modeling of a manipulator arm, and a Lagrange For-

malism for dynamic modeling of both a Multirotor

and arm . The system dynamics is decoupled in two

separate subsystems, one concerning the position of

the center of mass where a PID controller is used to

keep a stability of the whole system during hovering,

a second subsystem concerning the attitude of the ma-

nipulator arm, which the sliding mode controller is

used to control each link for a path trajectory between

a start point to a desired position of the end-effector.

That controller is modiﬁed and PD terms are added

for the sliding mode blocs in order to improve the out-

put signal and deﬁned good stability of the Q-PRR.

Simulation works are presented which validate and

show the efﬁciency of the proposed approach. Future

works will present an another approach for a dynamic

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

488

model by using a SimMechanics and VRML environ-

ment (Bouzgou et al., 2014) (Bouzgou et al., 2015).

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