A Motion Control

of Two-wheels Driven Mobile Manipulator

for Human-Robot Cooperative Transportation
Kohei Nozaki

Toshiyuki Murakami

Department of System Design Engineering

Keio University
Yokohama, Japan
Email: nozaki@sum.sd.keio.ac.jp

Department of System Design Engineering

Keio University
Yokohama, Japan
Email: mura@sd.keio.ac.jp

AbstractThis paper proposes a control method for humanrobot cooperative transportation using two-wheels driven mobile
manipulator. In the past, most of the robots which aim at
cooperation with human had three or more wheels, and there
is a possibility that their mobility is insufficient for human environment. Two-wheels driven mobile manipulator has excellent
mobility and can realize rapid and smooth motion such as turning
instantly. It has different motion characteristics depending on
the configuration, because of the nonholonomic constraint of
the mobile platform. And, there is a trade-off between the
robots mobility and the robots stability. In this paper, to the
above mentioned problem, we propose a method of configuration
control of the mobile platform by setting the manipulability
measure as a performance index. In the proposed method, by
maximizing or minimizing the performance index depending on
the situation, mobility and stability of the mobile platform in the
cooperative transportation is controlled.

I. I NTRODUCTION
With the development of robotic technology, robots are
widely used in various fields. As the population ages and birth
rates decline, many people are expecting robots to assist them
in their daily lives, such as in homes, offices, and hospitals.
So far, much research has been done on the robots which
work in cooperation with humans. Ikeura et al. have proposed
a control method for cooperation with a human based on
human impedance characteristics[1]. Al-Jarrah et al. proposed
a variable compliance control as a coordination mechanism
for a human and a robot[2]. While these studies consider the
fixed manipulator, there are several research considering the
mobile manipulator. Kosuge et al. developed an impedance
based motion control algorithm for the mobile robot with dual
arms[3]. Yamanaka et al. proposed a strategy of the virtual
impedance which determines the cooperative motion between
a human and a robot[4].
Most of the past research targeted mobile manipulators
which have three or more wheels. However, there is a possibility that their mobility is insufficient for human environment,
since human environment is complex and dynamic, such as
the presence of steps, slopes, narrow paths, and unknown
obstacles. If those robots change direction in a narrow space,
they need to make a three-point turn, and they are not

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adaptable in a narrow space. In addition, if they have some

casters, it is difficult to pass over the steps.
In this paper, we propose a control method for human-robot
cooperative transportation using two-wheels driven mobile
manipulator. Two-wheels driven mobile manipulator consists
of a mobile platform with two wheels and a manipulator with
6 joints and 5 links as shown in Fig. 2. It can achieve smooth
and rapid motion such as turning instantly. Furthermore, since
it doesnt have any casters, there is not so much difficulty to
pass over the steps. Therefore, it is suitable for tasks in human
living environment.
Cooperative transportation can be divided into some phases,
such as approaching to the object, lifting up/down of the
object, transporting, and positioning. The different control
law is needed in each phase. In this research, we restrict
our attention to the transporting task. The objective of the
cooperative transportation is that a robot and a human share
the load, and the robot moves according to the applied force.
We assume that the robot and human have interaction each
other only through the handling object. Therefore, the robot
can be seen as a blindfolded person.
Two-wheels driven mobile manipulator has different motion
characteristics depending on the configuration, because of the
nonholonomic constraint of the mobile platform. As shown in
Fig. 1, when the moving directions of the mobile platform
and the handling object are identical, the motion of the
mobile platform has much effect on the handling object. In
this configuration, the mobility of the mobile platform is
high, however, the disturbance from the handling object could
soon make the robot unstable. On the other hand, when the
above mentioned directions are orthogonal, the motion of the

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Fig. 1.

Motion characteristics and configuration of the mobile platform

Fig. 2.

Two-wheels driven mobile manipulator

Fig. 3.

Model of the manipulator

Fig. 4.

Model of the mobile platform

TABLE I

mobile platform has little effect on the handling object. In this

configuration, the mobile platform cant move in the direction
of the handling object, however, the disturbance from the
handling object doesnt cause a significant effect on the robot
stability. Thus, two-wheels driven mobile manipulator has a
trade-off between the mobility and the stability.
In this paper, to the above mentioned problem, we propose
a method of configuration control of the mobile platform by
setting the manipulability measure as a performance index.
In the proposed method, by maximizing or minimizing the
performance index depending on the situation, mobility and
stability of the mobile platform in the cooperative transportation are controlled.
II. M ODELING
In this section, kinematics and dynamics of the two-wheels
driven mobile manipulator are derived. Additionally, the simplified double inverted pendulum model, which is the equivalent model of the two-wheels driven mobile manipulator, is
introduced. The simplified double inverted pendulum model
is used in the stabilization control of the body of the mobile
platform in section III.
A. Two-wheels driven mobile manipulator

world coordinates
robot coordinates
radius of wheels
link length(i = 0, 1, , 5)
joint angle of manipulator(i = 0, 1, , 6)
rotation angle of right wheel
rotation angle of left wheel
direction angle of the mobile platform
tread of the mobile platform
position of the tip of the 3rd link in R
position of the mobile platform in W

m
m
rad
rad
rad
rad
m
m
m

and the world coordinates are assigned as shown in Fig. 3 and

Fig. 4.
1) Kinematics of the manipulator: The model of the manipulator is shown in Fig. 3. The position of the tip of the 3rd
link R x3rd in the robot coordinates and joint angles of the
manipulator q m are determined as eqs. (1) and (2). Note that
the passive joint is included in the manipulator.
R

x3rd
qm

=
=

[R x3rd ,R y3rd ,R z3rd ]T

[q0 , q1 , q2 , q3 , q4 ]T

(1)
(2)

The position of the tip of the 3rd link R x3rd and the velocity
x 3rd in the robot coordinates are obtained as follows.

Two-wheels driven mobile manipulator used in this paper is

shown in Fig. 2. This robot consists of a mobile platform with
two wheels, and a manipulator with 6 joints and 5 links. Since
the body of the mobile platform is not actuated, the rotating
body is considered as link 0 which has a passive joint. In this
paper, the manipulator and the mobile platform are modeled
separately. We assume that the handling object is fixed to the
end-effector, hence the posture angle of the handling object
can be obtained by the rotation angle of joint 5 and joint 6.
In this paper, we use joint 5 and joint 6 as rotary encoders
to detect the posture angle of the handling object. In order
to rotate these joints freely according to the posture of the
handling object, we treat them as passive joints. Therefore,
the control point is set to the tip of the 3rd link, not to the
end-effector. The parameters of the two-wheels driven mobile
manipulator are as shown in Table I. The robot coordinates

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PARAMETERS OF THE MOBILE MANIPULATOR

W
R
R
li
qi
r
l

W
Rx
3rd
Wx
p

R
R

x3rd
x 3rd

p5,i

= f (q m )
= Jm q m
R
z 0 R p5,0
=
=


z 4 R p5,4 q m

p5 R pi

(3)
(4)
(5)
(6)

where R pi is the vector from the origin of R to the origin

of i .
2) Inverse kinematics of the manipulator: The manipulator
has redundancy. In a redundant system, inverse kinematics is
obtained using pseudo-inverse matrix [7]. Inverse kinematics
of the manipulator is shown as eqs. (7) and (8).

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m
q
+
JmW

+ R
+
3rd + (I JmW
= JmW
J)
x
T
T 1
= W 1 Jm
(Jm W 1 Jm
)

(7)
(8)

where W is the weighting matrix. In eq. (7), the second term

in the right hand side is called null space. Null space is
utilized to control the joint space motion without interrupting
the workspace motion.
3) Kinematics of the mobile platform: Fig. 4 shows the
model of the mobile platform. The position of the mobile
platform in the world coordinates is determined as follows.
W

xp = [W xp ,W yp , ]T

(9)

By assuming that both right and left wheels do not slip, eqs.
(10) and (11) are obtained.
W

2
W

y p sin
2

x p cos +W y p sin +

= Rr

(10)

x p cos +W

= Rl

(11)

From eqs. (10) and (11), the following equations are obtained.
W

x p cos +W y p sin =

R
(r + l )
2

R
(r l )
=
W

(12)
(13)

Translational velocity of the mobile platform vp is given as

eq. (14).
vp =W x p cos +W y p sin
(14)
From eqs. (12) and (14), the following equation is derived.
vp =

R
(r + l )
2

(15)

4) Dynamics of the mobile manipulator: Using the NewtonEuler method, the dynamics of the mobile manipulator is
obtained as follows.

= M (q) + g(q) + h(q, q)

(16)

where, M , g and h denote the inertia matrix, the vector of

gravity and the vector of Coriolis and centrifugal force.
B. Simplified double inverted pendulum model
In this paper, we consider the simplified model, which is the
projected model of the mobile manipulator onto the saggital
plane. In this simplified double inverted pendulum model,
the body of the mobile platform is considered as the first
pendulum, and the line segment between the tip of the first
pendulum and the COG (center of gravity) of the manipulator
is considered as the second pendulum. Fig. 5 shows the model
of the simplified double inverted pendulum. The parameters of
the simplified double inverted pendulum model are as shown
in Table II.
TABLE II
PARAMETERS OF THE SIMPLIFIED DOUBLE INVERTED PENDULUM MODEL
Rx
G
l0

q0
lG
qG

m
m
rad
m
rad

position of the COG of the manipulator in R

link length of the first pendulum (body of mobile platform)
joint angle of the first pendulum (body of mobile platform)
link length of the second pendulum
joint angle of the second pendulum in the saggital plane

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Fig. 5.

Model of the simplified double inverted pendulum

COG position of the manipulator in the robot coordinates

is obtained as follows.

mi r i
R
xG = i
(i = 1, 2, , 5)
(17)
i mi
where mi denotes the mass of link i, and r i denotes the vector
to the tip of link i.
The link length lG and joint angle of the second pendulum
qG in the saggital plane are obtained as follows.

lG =
(R xG l0 sin q0 )2 + (R zG l0 cos q0 )2 (18)
R

zG l0 cos q0
1
qG = cos
(19)
q0
lG
Using Lagrangian dynamics, the nonlinear equations of
motion are derived. By linearizing them in the neighborhood
of q0 = 0 and qG = 0, following linearized dynamics of the
simplified double inverted pendulum is obtained.

r
r
l
l


(20)
M
q0 + g = 0
qG
qG
where, M  and g  denote the inertia matrix, and the vector of
gravity.
III. C ONTROL S YSTEM
In this paper, the manipulator and the mobile platform
are controlled separately. The manipulator has redundancy. In
the manipulator, as a primary task, impedance control of the
tip of the 3rd link is conducted. At the same time, posture
control of the manipulator is conducted in the null space of
the manipulator as a secondary task. In the mobile platform,
configuration control of the mobile platform, and the attitude
stabilization control of the body of the mobile platform are
conducted at the same time.
A. Control of the manipulator part
1) Impedance control of the tip of the 3rd link: Humans
intentional force includes some factors of disturbance, and
this could make the robot unstable. Therefore, the compliant
motion is required for the robot. As a primary task of the

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Fig. 6.

Fig. 7.

2link mobile manipulator model in the horizontal plane

manipulator, impedance control of the tip of the 3rd link is

conducted. Humans intentional force F hum is estimated by
using the reaction torque estimation observer[10].
Let us set the apparent impedance characteristic as follows.

x3rd + De x 3rd + Ke x3rd = F

Me

(21)

Here, x3rd denotes the displacement of the tip of the 3rd

link in the arbitrary direction where the humans intentional
force is added.
The acceleration reference can be obtained as follows.
R

1
R
) (22)
ref
3rd +Ke R x3rd + F
x
3rd = Me (De x

2) Posture control of the manipulator: The direction angle

of the handling object is detected using joint 5. Therefore,
the posture of the 3rd link 3rd should be kept parallel to
the horizontal plane. The posture control is conducted by
controlling the angle of joint 4 based on PD control as follows.


res
res
(23)
q4refnull = Kp
3rd +Kv (0.0 3rd )
2
res
res
res
res
res
3rd = q0 + q2 + q3 + q4
(24)
B. Configuration control of the mobile platform
Two-wheels driven mobile manipulator has different motion
characteristics depending on the configuration, because of the
nonholonomic constraint of the mobile platform. There is a
trade-off between the robots mobility and the robots stability.
In this paper, to the above mentioned problem, configuration
control of the mobile platform setting the manipulability
measure as a performance index is proposed.
1) Velocity contribution of the mobile platform to the handling object: We investigate the contribution of the mobile
platform to the manipulability of the handling object, which
varies with the configuration of the mobile platform to the handling object. For simplicity, manipulator is considered as two
link model (arm and wrist) in the horizontal plane, as shown in
Fig. 6. The velocity relation between the mobile platform and
the end-effector is described as eq. (25). Here, the velocity of
the end-effector is considered in the moving direction of the
mobile platform and the perpendicular direction.
m

= J  w

(25)

=
=

[m x,m y]T
[ , ]T
r l


J11 J12
=


J21
J22

(26)
(27)

w
J

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Manipulability measure, det(J  J T )

R
R

{L1 sin q1 + L2 sin(q1 q5 )} (29)

2
W
R
R

+
{L1 sin q1 + L2 sin(q1 q5 )} (30)
=
J12
2
W
R

{L1 cos q1 + L2 cos(q1 q5 )}
=
(31)
J21
W
R

= {L1 cos q1 + L2 cos(q1 q5 )}
(32)
J22
W
2) Manipulability measure: We set manipulability measure
as a performance index which indicates the contribution of
the motion of the mobile platform to the manipulability of
the handling object. Manipulability measure is represented as
follows using Jacobian matrix J  .

J11

V (q1 , q5 ) = det(J  J T )

(33)

It varies with the configuration of the mobile platform to the

handling object as shown in Fig. 7. From the result, when the
direction of the handling object and the moving direction of
the mobile platform are identical, manipulability measure is
large, and the motion of the mobile platform has much effect
on the handling object. On the other hand, when the above
mentioned directions are orthogonal, manipulability measure
is small, and the motion of the mobile platform has little effect
on the handling object.
3) Configuration control of the mobile platform based on
the performance index: In eq. (33), q1 can be regarded as
nearly constant, because the impedance control of the tip of
the 3rd link is conducted. Assuming that a human keeps the
posture of the handling object as he/she wants, by controlling
the configuration of the mobile platform so that the angle of
the wrist q5 minimizes or maximizes performance index V ,
the contribution of the motion of the mobile platform to the
manipulability of the handling object can be changed. The
reference for the configuration control of the mobile platform
is given as follows.
V
Kv res
ref = k Kp
q5

(34)

4) Design of the switching function: In eq. (34), k is a

switching function which determines whether the motion of
the mobile platform should be restricted or enhanced. In this
paper, we design the switching function as follows.

(28)

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k=

1 exp((v va ))
1 + exp((v va ))

(35)

Fig. 8.

Switching function

When the velocity of the handling object is larger than the

constant velocity va , we judge that the robot is in moving
phase. Then, the performance index is minimized, and the
mobile platform approaches to the configuration where the
motion is enhanced, and vice versa. The sign of k for the
minimization of the performance index is opposite to the
usual one, because the direction of the mobile platform which
minimizes the performance index is opposite to that of joint
5. In this paper, switching function k is only related to the
velocity of the handling object, but the design of the switching
function for better performance should be considered in the
future work.

Fig. 9.

TABLE III
P HYSICAL PARAMETERS IN SIMULATION
link length of link0
l0
0.20 m
link length of link1
l1
0.30 m
link length of link2
l2
0.30 m
link length of link3
l3
0.10 m
link length of link4
l4
0.10 m
link length of link5
l5
0.10 m
radius of wheels
R
0.20 m
mass of link0
m0
5.2 kg
mass of link1
m1
1.35 kg
mass of link2
m2
1.35 kg
mass of link3
m3
0.5 kg
mass of link4
m4
0.5 kg
mass of link5
m5
0.5 kg
mass of right(left) wheel
m(r,l)
4.2 kg

C. Stabilization control of the body of the mobile platform

The attitude of two-wheels driven mobile manipulator is
unstable and some control is required to keep its stable
attitude. Stabilization control is considered in the saggital
plane using the simplified double inverted pendulum model.
From the dynamics of the simplified double inverted pendulum model in eq. (20), the motion equation of the body of
the mobile platform is obtained as follows.




q0 + M0G
qG + g0 = 0
r + M0l
l + M00
M0r

Then, the sum of the acceleration reference of both wheels for

attitude stabilization is obtained as follows.
1

Kp(r,l) (q0res q0cmd )
rref+ lref =  {M00
M0r


+ M00
Kv(r,l) (q0res q0cmd )M0G
qG g0 }(38)
D. Whole Control System
The whole block diagram is shown in Fig. 9.
IV. S IMULATION
In order to verify the validity of the proposed method,
simulation was conducted.
As a human input, the following force was inputted to the
robot by using impedance model.


x + Dhum x + K hum x (39)
F hum = M hum

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TABLE IV
C ONTROL PARAMETERS IN SIMULATION
proportional gain in posture control
Kp
900.0
derivative gain in posture control
Kv
60.0
proportional gain of wheel
Kp(r,l)
900.0
derivative gain of wheel
Kv(r,l)
60.0
proportional gain in configuration control
Kp
1.0 106
derivative gain in configuration control
Kv
90.0
disturbance observer gain
gDOB
30.0
reaction torque estimation observer gain
gROB
100.0
parameter in switching function

300.0
threshold velocity
va
0.06

(36)



and M0l
are equal because the simplified model
Here, M0r
is considered in the saggital plane. Therefore, from eq. (36),
following equation is obtained.
1


q0 =  {M0r
(r + l ) + M0G
qG + g0 }
(37)
M00

The whole block diagram

Here, human impedance was set as below based on the

research by T. Tsuji et al. (1995) [13].
(Mxhum , Dxhum , Kxhum )

( 1.18, 29.09, 474.26)

(Myhum , Dyhum , Kyhum )

( 1.08, 6.45, 58.52)

As the gravity added to the handling object, force with

magnitude 9.8N was added to the end-effector.
Physical parameters and control parameters used in simulation were as shown in Table. III and Table. IV. Impedance
characteristic used in impedance control of the tip of the 3rd
link was set as below.
(Me , De , Ke ) = ( 1.0, 20.0, 100.0)
In simulation, the behavior of the mobile manipulator is
examined when the desired trajectory was set as shown in
Figs. 10 and 11.

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Fig. 10.

Position in

Fig. 13.

WX

axis

Direction angle

Fig. 11.

Fig. 14.

Velocity and switching function

Human force

Performance index V (q1 , q5 )

The simulation results are shown from Fig. 10 to Fig.

14. From Fig. 11, we can see that the switching function is
changed as the translational velocity of the handling object
changed. From Fig. 13, it is found that the configuration of the
mobile platform changed from easy to move configuration to
hard to move configuration, as the switching function changed.
As a result, the effect of the motion of the mobile platform
to the handling object became very small as shown in Fig.
14. From the results, we can see that the configuration control
according to the condition of the task was succeeded. However,
this is because the human sustains the robot to keep the
position of the control point constant. As a result, it becomes
a burden to a human as shown in Fig. 12. As for future works,
how to change configuration while avoiding the burden to a
human should be considered.

[3]

[4]

[5]

[6]

[7]
[8]

V. C ONCLUSION
In this paper, a control method for human-robot cooperative
transportation using two-wheels driven mobile manipulator
was proposed. In order to change the contribution of the
motion of the mobile platform to the handling object according
to the condition of the task, configuration control of the mobile
platform based on the performance index was proposed. The
validity of the proposed method was verified by simulation.
As for future works, following work should be done.
Consideration how to change configuration while avoiding the burden to a human.
Design of the switching function for better performance.

[9]
[10]

[11]
[12]
[13]

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