Adaptive Control of Mobile Manipulator Robot based on Virtual

Decomposition Approach

Abdelkrim Brahmi

, Maarouf Saad

, Guy Gauthier

, Wen-Hong Zhu

and Jawhar Ghommam

Electrical Engineering Department, École de Technologie Supérieure, Montréal, QC, Canada

Space Exploration Canadian Space Agency,

Montréal, QC, Canada

Ecole National d’Ingenieurs, Research Unit on Mechatronics and Automation Systems, Sfax, Tunisia

Keywords: Virtual Decomposition Control, Adaptive Control, Centralized/Decentralized Control, Mobile Manipulator

Robot.

Abstract: This paper presents an adaptive control scheme for a mobile manipulator robot based on the virtual

decomposition control (VDC). The control strategy was tested on three degrees of freedom manipulator arm

mounted on two degrees of freedom mobile platform to track a desired trajectory. The desired trajectory

is obtained from the workspace trajectory using the inverse kinematics. Differently to the known

decentralized control that divides the mobile manipulator into two subsystems, in this paper, the mobile

manipulator has N degrees of freedom, divided virtually into N subsystems. The applicability of the

proposed scheme is demonstrated in real time validation. The experimental results show the effectiveness of

the VDC approach.

1 INTRODUCTION

The need for robots capable of locomotion and

manipulation has led to the design of mobile

manipulator robot (MMR) platforms. A mobile

manipulator is a robotic manipulator arm mounted

on a mobile platform. Typical examples of MMR

include satellite arms, underwater robots in seabed

exploration and vehicles used in extra-planetary

exploration. The mobile manipulator comprises two

subsystems, that is, the mobile platform and the

manipulator arm subsystem.

However, this

significantly complicates the robotic system as its

control design complexity increases greatly. The

problem of controlling the mechanical system lies in

the fact that it imposes a set of kinematic constraints

on the coordination of the position and velocity of

the mobile manipulator. Few works have been

proposed to solve the control problem of these

robotic systems, which have high degrees of

freedom and are tightly interconnected.

In the recent years, the research in the design,

control, stability, and path tracking of mobile

manipulators have significantly increased. Most of

these studies have thus far focused on the tracking

control of mobile manipulators. Two main schemes

of control are developed in the literature: The first

one is centralized control, in which the mobile

manipulator is regarded as one system and the

controller is designed for the full system. The second

one is the decentralized control, in which controllers

for two subsystems are designed separately and no

coupling is considered. In the first approach, where

the mobile platform and the manipulator arm are

regarded as a complete system, many control

approaches were developed and implemented. In

(Yamamoto and Yun 1994), the authors focused on

the interaction between the manipulator arm and the

mobile platform. A nonlinear feedback control was

developed to compensate the dynamic interaction.

Studies discussing the problem of modeling and

control of mobile manipulator were given in (Chung

and Velinsky 1998, Seraji 1998, Song et al. 2005,

Aviles et al. 2012, Galicki 2012), where the robotics

system is considered as a complete system. Many

other works have used decentralized control for this

type of robotic systems as in (Tan and Xi 2001, Ngo

et al. 2007, Ge et al. 2008, Chen et al. 2015) where

the manipulator arm and the mobile platform are

viewed as two separated subsystems. LQR controller

for mobile manipulator was proposed (Chen et al.

2015), where the manipulator arm and the mobile

254

Brahmi, A., Saad, M., Gauthier, G., Zhu, W-H. and Ghommam, J.

Adaptive Control of Mobile Manipulator Robot based on Virtual Decomposition Approach.

DOI: 10.5220/0005975402540261

In Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2016) - Volume 2, pages 254-261

ISBN: 978-989-758-198-4

platform are controlled independently. In (Ge et al.

2008) a sliding mode control was proposed to

control the mobile platform and a nonsingular

terminal sliding mode control for the manipulator's

arm.

Conventional control approaches consider

integrated mobile manipulator dynamics. However,

in practice, it becomes very difficult to get the exact

model and uncertainties may still exist. In (Wu and

Sun 2014) an adaptive tracking control scheme was

proposed for a mobile manipulator with the presence

of uncertainties and disturbance based on suitable

reduced dynamic model. An adaptive sliding mode

controller based on the backstepping applied to the

trajectory tracking of the wheeled mobile

manipulator was introduced in (Chen et al. 2013).

An adaptive control scheme based on suitable

reduced dynamic model was proposed in (Dong

2002), without considering any disturbance. To

overcome the problem of dynamic modeling and

dynamic control, some researchers proposed

adaptive control based on neural network control

and fuzzy logic approaches. For instance, non-

model-based techniques have been developed for a

different type of mobile manipulator robot with

dynamic parameters uncertainties (Mai and Wang

2014, Peng et al. 2014, Wu et al. 2014).

A. Main Contribution

All previous studies based on Lagrangian or

Newton/Euler approaches require knowledge of the

exact parameters of the system. In practice, this is

difficult, and the obtained model is usually

uncertain. For these types of systems with large

degrees of freedom, and which are tightly coupled,

adapting the parameters using methods based on full

dynamics is very complicated due to the huge

number of parameters involved.

To overcome, this problem we propose in this

paper a novel adaptive decentralized approach based

on an extension of the virtual decomposition control

(VDC) methodology in ( Zhu 2010, Brahmi et al.

2013) originally designed for fixed-base robotic

systems with large degrees of freedom. Some of the

many advantages of this approach are: 1) the whole

dynamics of the system can be easily found based on

the individual dynamics of each subsystem; 2) the

control only uses subsystem dynamics while

guaranteeing the stability of the entire system; and

3) it makes the adaptation of the physical parameters

very simple and systematic.

In opposite to known decentralized control

techniques that divide the mobile manipulator into

two subsystems, in our work, if the mobile

manipulator that has N degrees of freedom then the

robotics system is divided virtually into N

subsystems. This simplifies the control and the

dynamic parameters adaptation.

The rest of the paper is organized as follows.

Section 2 presents the modeling of the system while

section 3 presents the control problem statement.

Section 4 explains the control design and

experimental results are given in section 5. Finally, a

conclusion is given in section 6.

2 MODELING AND SYSTEM

DESCRIPTION

Before Before giving the rationale behind the virtual

decomposition approach, we start by giving a brief

formulation of the kinematic and dynamic modeling

of the mobile manipulator robot under consideration.

Figure 1 shows the holonomic manipulator arm

mounted on nonholonomic mobile platform, where

the manipulator has p-DOF, the mobile platform has

m-DOF and the full robotic system has n=m+p-

DOF.

Figure 1 shows the MMR with

being the

position/orientation vector of the MMR end–

effector.

Figure 1: n- DOF mobile manipulator robot.

2.1 Kinematics

The relationship between the end effector velocity

∈ \

of the mobile manipulator and the

Adaptive Control of Mobile Manipulator Robot based on Virtual Decomposition Approach

255

linear/angular velocity of mobile platform

qxy

⎡⎤

⎣⎦





and joints velocities of the

manipulator arm

[]

,..,

qqq=



is given by:



VJq

(1)

where,

[, ]

qqq=∈



and

6×

∈ \

is the

Jacobian matrix.

2.2 The Mobile Manipulator Dynamics

The dynamic model of the mobile manipulator

developed using Lagrangian approach is given in the

general joint space by the following equation:

() ( ) ()

qq Cqqq Gq A

τλ

++=Ε+

  

(2)

where

∈ \

is the mass matrix,

∈ \

represents the Coriolis and centrifugal terms,

∈ \

is the vector of gravity,

,, ∈



qqq

are the

generalized coordinate vector, the joint velocity and

the acceleration vectors respectively ,

∈ \

is a

full rank transformation matrix and

∈ \

is the

input control vector,

∈\

is the constraint

matrix and

∈ \

is the constraint force.

The mobile manipulator robot is considered as a

fully actuated arm mounted on the nonholonomic

mobile platform. This nonholonomic constraint is

given as:

() 0Aqq=



(3)

The objective is to eliminate the term of

constraint

, using the kinematic equations of the

mobile platform and the right and the left wheels’

angular velocities



and



that are obtained by the

following expression:

()qSq



(4)

where,

[, ]

qqq=



is the generalized coordinate

linear/angular velocities,

()Sq

is in the null space of

the kinematic constraint matrix

()

and

[,,]

RLa

qqq





are the generalized coordinate

angular right/left wheels velocities and the

manipulator joint velocities. Therefore, we obtain:

() () 0

SqAq=

(5)

Based on (2), (4) and (5) the dynamic model (2)

can be expressed by a linear relationship of the form:

() ( ) () ( )

,,,,qqqqWqq

ηηητ

Η+Β +ℑ= Γ=

   



(6)

where

()

qSMSΗ=

()

qSMSCS=+



()

qSGℑ=

The physical parameters of the robot

are unknown and had to be estimated. Using the

estimate parameters of

, noted

,Γ

the equation of

parametrization expression can be written as:

() ( ) () ( )

ˆˆ ˆ

,,,,qqqqWqq

ηη ηη

Η+Β +ℑ= Γ

   



(7)

In the classical decentralized/centralized

approach based adaptive control, it’s very difficult to

obtain these parameters when the degree of freedom

increases. Usually, the size of the parameters vector

Γ can be greater than 100 in this category of robots (

Zhu 2010, Brahmi et al. 2013). As a solution of this

serious problem, we use a novel adaptive control

based on the virtual decomposition approach where

the problem of control and adaptation of parameters

are converted to each rigid body and each joint.

To simplify the control formulation, the

following assumption is made:

Assumption 2.1: All the joints’ velocities of the

mobile manipulator robots are available for

feedback.

The dynamics given in equation (6) has the

following properties:

Property 2.1: the matrix H is symmetric positive

definite.

Property 2.2: the matrix

2QH B=−



is skew

symmetric, that is, for any vector x, we have:

()

xH Bx−=



3 CONTROL PROBLEM

STATEMENT

The idea of the VDC is to break down the robotic

system into a graph consisting of several objects and

open chains. An object is a rigid body and open

chain consists of a series of rigid links connected

one by one by a hinge and has a certain degree of

freedom. The dynamic coupling between the

subsystems can be represented by the flow of virtual

power (FVP) at the cutting point. This refers to the

principle of virtual work (Zhu 2010, Brahmi et al.

2013) and results in one open chain.

For the controller design, only the dynamics of

the rigid bodies and the joints are required.

The control objective is to generate a set of

torque inputs such that the joint position’s tracking

error converges asymptotically to zero. Formally

speaking, the control problem is to design the

control input:

ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics

256

()



such that the following limits hold:

lim 0, lim 0

qq qq

→∞ →∞

−= −=



where,

q ∈ \

∈ \

are the measured and

desired joint angular position and velocity of the

mobile manipulator.

4 CONTROL DESIGN

The overall control system is designed using the

following steps:

- The joint space desired trajectories are obtained

from the workspace desired trajectories using the

inverse kinematics(1).

- The required joint space trajectory are computed

from the desired trajectory in joint space.

- The required velocity

∈ \

of the n body-

fixed frames

B illustrated in Figure 2. is calculated.

- The VDC approach is used to simplify the

problem of adaptation of the parameters of the

complete system, where this problem is converted

into a problem of estimation of the parameters of

each subsystem. From the velocities computed in the

first step, the estimated parameters are calculated.

- The control law of the mobile manipulator robot is

finally designed.

4.1 Design

Step 1: the desired joint velocity

∈



calculated based on (1), and then the required joint

velocities are by



()

dpd

qqKqq=+ −



(8)

with

is a positive gain matrix.

Step 2: In this step, the goal is to virtually

decompose ( Zhu 2010, Brahmi et al. 2013) the

robotic system into several parts and open chain

elements. Each part is a rigid body and open chain

consists of a series of rigid links connected one by

one. This decomposition is illustrated in Figure 2.

Figure 2: Virtual decomposition of the MMR.

In this step the linear/angular velocity vectors of

each frame B

is defined as:

,..., , , ,...,

BnvBB

VqqVVV

⎡

⎤

⎣

⎦



and is calculated as

follows:

BjBB

Vzq UV



(9)

where,



is the joint velocity,

V ∈ \

the linear

and angular velocity of the corresponding frame,

V ∈ \

is the velocity of the mobile platform and z

is defined as

[]

0 0 1000=

for prismatic axes and

[]

0 0 0001=

for revolute axes.

The transformation matrix of force/moment

vectors from frame B to frame A is defined by:

()

⎡

⎤

⎢

⎥

⎢

⎥

⎣

⎦

AA A

BB B

Sr R R

(10)

where

33×

∈ \

is the rotation matrix between

frames A and B, and

()

33×

∈ \

is a skew

symmetric matrix built from the vector

33×

∈ \

linking the origins of frames A and B, expressed in

the coordinates of frame A.

The velocity vector is defined as follows:

,..., , , ,...,

rrr rT rT

BnvBB

VqqVV V

⎡

⎤

⎣

⎦



where



are the joint velocities, and the

∈

vectors represent the velocity of each frame Bj.

The following relates the velocity propagation

along the structure:

Adaptive Control of Mobile Manipulator Robot based on Virtual Decomposition Approach

257



rr Tr

BjBB

Vzq UV

(11)

In general, we can write the system in a matrix

form by using the Jacobian matrix:



VJq

(12)

with

,..., , , ,...,

rrrTrT

BnvBB

VqqVV V

⎡⎤

⎣⎦



and

7 ×

∈ \

being the VDC Jacobian matrix of the system.

The dynamics of the j-th rigid body is given in

the linear form by the following equation:

jjjjj

BBBBBBj

vvvvvv

MV CV G

F MVCVG

⎧

=++

⎪

⎨

⎪

=++

⎩



(13)

with

∈ \

being the matrix of inertial

term,

∈ \

the matrix of

centrifugal/Coriolis term,

GG∈ \

the vector

related to the gravity. The vector of resulting forces /

moments acting on the rigid body is computed by an

iterative process as follows:

11 2 2 3 3

22 22

11 11

nn nn

BB BB

vv v

vv TT TT BB

BB TT

FF UF

FUFUFUF

FF UF

−

−−

== + −

(14)

The dynamic model of the j-th rigid link (13)

based on its required velocity is given by:

jj jj j jj

rrr

BBBBBBBB

rrr

vvvvvvvv

F MVCVGY

⎧

=++=

⎪

⎨

⎪

=++=

⎩



(15)

where,

θθ

∈

is the parameters’ vector of the

j-th rigid link, and

613

∈ \

the dynamic regre

s sor matrix.

Since the physical parameters of the j-th rigid

body and the mobile platform are unknown and need

to be estimated, then the estimated vectors

ˆˆ

θθ

are used and the required force/moment is obtained

as follows:

()

jj j j j

BBB BBB

vvvvvv

FY KVV

⎧

=+ −

⎪

⎨

⎪

=+ −

⎩

(16)

where,

ˆˆ

jjj

Bj B B B v v v v

Ys Ys

θρ θρ



are the

adaptation functions, and are chosen to ensure

system stability,

()

jjj

BBBvvv

VVsVV=− =−

and

ρρ

Bj v

are positive gains.

The vector of the required resulting

forces/moments acting on the

j-th rigid body is given

by an iterative process ( Zhu 2010, Brahmi et al.

2013) as in (14).

The dynamics of the

j-th joint actuator is

expressed by the following equation:

()

τξ

 

aj m j j j

qqq

(17)

where,

()



represents the friction and

gravitation force / torque terms and

is the

moment of inertia of the motor driving this joint.

The dynamics (17) based on its required velocity is

expressed in the linear form by the following:

()

rrrr

aj m j j j aj aj

Jq qq Y

=+ =

 

(18)

where,

∈ \

is the parameters’ vector of the j-th

joint actuator, and

∈ \

is the dynamic

regressor (row) vector, defined in ( Zhu 2010,

Brahmi et al. 2013). Since the physical parameters

of the

j-th actuator are unknown and need to be

estimated, then the estimated vector

is used and

the dynamic (18) becomes:

()

aj aj aj aj j j

YKqq

τθ

=+ −



(19)

where,

jjj

aj a a a

θρ



is the adaptation function, and

is chosen to ensure system stability,

()

=−



arjj

and

are positive gains.

Finally, the input control torque at the

j-th mobile

manipulator’s joint is computed from the desired

torque obtained from (19)

and the required force

at cutting point

, identified

as:

rTr

aj B

ττ

(20)

Lemma 4.1: Consider the j-th rigid dynamics (13,

14) and the joint actuator dynamics (17), under the

control design (16, 19 and 20) and the boundedness

of the estimated parameters. The control objective is

satisfied and the error tracking states are

asymptotically stable.

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258

Proof: Consider the Lyapunov function candidate:

aj p

VV VV

=+ +

∑∑

(21)

where

V is a non-negative Lyapunov candidate

function related to the

j-th rigid link,

V is a non-

negative Lyapunov candidate function of the

j-th

joint and

V is a non-negative Lyapunov candidate

function of the mobile platform. These three

Lyapunov candidate functions are chosen as follow:

()()

(

)

()

()()

()

jj jjj

ji ji

aj aj

jBBBBB

aj m j j

pvvvvv

VVVMVV

VJqq

VVVMVV

θθ

−

⎧

⎡⎤

=− −

⎪

⎢⎥

⎣⎦

⎪

⎨

⎡⎤

⎪

=−+

⎢⎥

⎪

⎢⎥

⎪

⎣⎦

⎪

=− −+

⎪

⎩

∑

∑∑

∑





(22)

where

,,,,and

ji ji i i i i

B B v v aj aj

θ θ θθθ θ





are the i-th

elements of the corresponding vector parameters.

The first derivative of the Lyapunov candidate

function (21) is given as follows:

aj p

VV VV

=+ +

∑∑

 

(23)

By using the definition of the virtual power and

the choice of the parameter function adaptation as in

(16) and (19); it is straightforward to prove that



always decreasing and is given as follows:

()()

()()()

jj jjj

BB BBB

rr r

ajj vv vvv

VVVKVV

qq VV KVV

=− − −

−−−− −

∑





(24)

The stability analysis shows that



is always

decreasing and the system is asymptotically stable in

the sense of Lyapunov. The reader can found the

detailed proof stability in ( Zhu 2010).

5 EXPERIMENTAL RESULTS

Experimental implementation is carried out on a 3-

DOF MMR as illustrated in Figure 3. A Zigbee

technology communication is used between the

application program implemented in Simulink

Matlab® and the mobile manipulator robot.

The VDC

approach is implemented in real time using Real-Time

Workshop (RTW) of Mathworks

Figure 3 shows the

complete structure design of the control. This block

diagram consists of Zigbee communication, low level

(LL) controller (PI controller) and High level (HL)

controller (virtual decomposition control) and

measurement sensors (encoders).

Figure 3: Real-time setup.

The control strategy was tested on 3-DOF mobile

manipulator robot to track a desired trajectory in

Cartesian space presented in Figure 4. The two

wheels of the

j-th mobile manipulator robot platform

are actuated by two DC-motors HN-GH12-2217Y

(DC-12V-200RPM 30:1) and its angular positions

are given by using encoder sensors (E4P-100-079-

D-H-T-B). All joints of the arm manipulator are

actuated by Dynamixel motors (MX-64), this

actuated gives the measurement of different

parameter’s as the angular position, the angular

velocity, the torque and many others parameters

used in the control and analysis. In this experimental

test the Dynamixel motors are configured to be

controlled on torque mode and are linked to ATmega

32 via TTL level multi drop bus.

The starting point is

()

0.14, 0.02, 0.4214

P =−

the final point is

()

7.8, 0.28,0.4759

P =−

, without

end-effector orientation along X, Y or Z-axis and the

actual end-effector mobile manipulator robot

position is

()

0.18, 0.045, 0.3565

P =−

. The inverse

Adaptive Control of Mobile Manipulator Robot based on Virtual Decomposition Approach

259

kinematics is used to transform the trajectory

generated in the workspace to the joints space. The

controller gains are chosen as follows:

()

21.2,8,130,3.7,3.7 , 0.8, 0.7, 0.8

pvBaj

KKKK====.

The sampling time is set to 0.015 seconds.

The trajectory tracking in the Cartesian space is

presented in Figure 4 and Figure 5 (a-b-c). It can be

seen a good position tracking from Figure 5 (a-b-c).

This good tracking is confirmed by the related errors

between the desired values and the real ones shown in

Figure 5 (d-e-f). All the input torques of the three

joints of the arm manipulator and that of the right/left

wheels are illustrated in Figure 6 (a-b-c-d-e).

Figure 4: Desired trajectory of manipulator’s end effector.

Figure 5: (a) Tracking trajectory of x-position, (b)

Tracking trajectory of y-position (c) Tracking trajectory of

z-position, (d) Tracking error of x-position, (e) Tracking

error of y-position (f) Tracking error of z- position.

Figure 6: (a) The input torque of the joint 1, (b) The input

torque of the joint 2 (c) The input torque of the joint 3, (d)

The input torque of the right wheel 1, (e) The input torque

of the left wheel.

6 CONCLUSIONS

In this paper, a decentralized control based on the

virtual decomposition strategy was presented to

control three degrees of freedom mobile manipulator

robot to track desired trajectories generated in

Cartesian space. The control law is designed based

on the virtual decomposition approach, and the

global stability of the systems is proven through the

virtual stability of each subsystem. The proposed

control design ensures that the workspace position

error converges to zero asymptotically. The

experimental results showed the effectiveness of this

control approach where the tracking errors of the

desired trajectory in workspace converge to zero.

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