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Control Passive Mobile Robots for Object Transportation - Braking Torque

Analysis and Motion Control

Conference Paper  in  Proceedings - IEEE International Conference on Robotics and Automation · April 2007
DOI: 10.1109/ROBOT.2007.363907 · Source: DBLP

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2007 IEEE International Conference on ThD4.1
Robotics and Automation
Roma, Italy, 10-14 April 2007

Control Passive Mobile Robots for Object Transportation

- Braking Torque Analysis and Motion Control -
ZhiDong Wang, Kenta Fukaya, Yasuhisa Hirata, and Kazuhiro Kosuge

Abstract— In this paper, we propose a concept on realizing

impedance-based motion control of passive type robots for
transporting a single object in coordination with a human
operator. In this research, we developed a prototype of passive
type robot referred to as Passive Robot Porter or PRP, which
consists of three omni-directional wheels with MR Brakes, and
on-board computer system. We analyze the singularity of PRP
braking torques, and control the brake torque of each wheel
based on the brake force/moment constraint so that impedance
characteristics is realized on PRP with human’s pushing. This
allows a PRP to track a 2D path which includes motion
perpendicular to human’s pushing direction without using servo
motors, and multi-PRPs to work cooperatively on handling
Fig.1 . PRP System for Object Transportation with Human
a single object with orientation control for avoiding collision
with obstacles. Experimental results are shown to illustrate the design of a passive robot, singularity analysis of the system
validity of the proposed concept.
and impedance-based motion control under braking torque
constraints. Finally, experimental results on impedance-based
I. I NTRODUCTION
motion control and cooperative object handling control are
Beyond the conventional industrial applications, re- shown to illustrate the validity of the proposed system.
searches and developments on robotics recently are more
focused on applications or areas related to our daily life, II. PASSIVE ROBOTICS
such as, entertainment, service, medical and welfare applica- Conventional robot systems are developed by incorporat-
tions, etc. With these applications, human-robot cooperative ing active actuators for both generating and controlling the
systems on object handling and transportation are required. motion of the system. Sometimes we have to design high
Studies in [1]∼[3] demonstrated various interesting and gear ratio for small actuators. This lets the system lose its
essential tasks on it. These studies address the dynamic backdrivability. In other cases, we need to install big motors
interaction between human and robots, which is one of the to keeping the backdrivability and to have enough power
most important factors on leading Robot Technology (RT) both. Either of the aforementioned systems will not be safe
into our daily life. if actuators are out of control for unexpected reasons. There-
As robots interact to our daily activities, several techni- fore, some safety measures should be carefully designed
cal issues are becoming more important. One of the most especially if the system is working in an environment with
important issues is safety pointed out by many researchers. human.
In general, robotic systems consist of active actuating com- Goswami’s group [4] proposed the concept Passive
ponents such as motors. We are generally concerned about Robotics, and the resulting system does not include any force
any problem on robot control which will lead to unexpected driving component for motion generation. The motions are
or undesirable motion of robots. This situation may let always due to the external forces applied to the system. In
robots hit and hurt human being. Many researchers have their system, some passive components such as mechanical
been developing methods to examine this kind of collisions springs and dampers are installed on the manipulator. By
or predict the possibility of collisions by installing various controlling the physical parameters of passive components,
sensors on the robot systems. These strategies are feasible various motions are realized with the external forces on
but with limitation, especially when the type and number of the end-effector. Unexpected motion due to out of control
sensors are not sufficient. will not happen in a passive system since motion is a
In this research, we focus on the development of a product of the applied force of the human operator. In this
passive type robot system, which does not include any active meaning, higher safety on motion control is achieved. In
component such as motor (Fig.1). This can realize a high addition, some passive type actuators such as MR brake have
level of safety without losing system’s performance on object good performance on torque-weight ratio. This is usually an
transportation. In this paper, we will address the system advantage on developing simple and low power consuming
Z.D. Wang is with the Department of Advanced Robotics, robotic systems. From these points, the concept of Passive
Chiba Institute of Technology, Narashino, Chiba, 275-0016, Japan. Robotics is unique and potential on achieving tasks with
zhidong.wang@it-chiba.ac.jp robot-human interaction.
K. Fukaya, Y. Hirata and K. Kosuge are with Department of Bioengineer-
ing and Robotics, Tohoku University, Sendai, 980-8579, Japan. {fukaya, Demonstrated by many researches, such as Davis and
hirata, kosuge}@irs.mech.tohoku.ac.jp Book’s work[5], passive actuators show its advantage on

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 ThD4.1

manipulator control. An earlier work on developing ob- IV. C ONTROL OF PASSIVE ACTUATOR
ject transportation system based on the concept of Passive Passive actuators have some unique controlling character-
Robotics is discussed in [6]. This system is designed by istics. The output of active actuators such as servo motor will
Peshkin and Colgate’s group. The system consists of motors affect the control target independent of the target’s motion.
for steering three passive wheels. The robot and object are However, the output of a servo-brake will not affect the
moved by the pushing force of the human operator, but motion of the target if it is not moving and will not move.
trajectory of the system is controlled by a steering wheel. This is a very important feature in realizing safety actions.
In our group, as well as active working helper system [7], Also some constraints make the control of passive type robot
we have developed a passive working helper [8][9] and it systems to be different from controlling ordinary systems.
consists of two MR brakes. This system assists user to Let us consider how the output torque of an actuator is
walk in environments with obstacles and it also provides applied to a mobile robot in the case of active and passive
proper dynamic characteristics to prevent the user from actuator such as motor and servo-brake respectively.
loosing his footing. However, the system is using normal
(i) Motor Output Torque
wheels with non-holonomic constraints. Cobot also consist
of active type actuator even it is for passive application. It is well known that the torque applied to the wheel
Recently, Ryu and Pathak’s group also proposed a passive will be equal to the output torque of the motor as,
based control law for differentially driven mobile robot [11]. τm = km Im (1)
In this research, we focus on developing a robot system Im denotes the control input of a motor and km
consisting of passive actuators and investigate the dynamic denotes the torque constant of the motor. Without
characteristics and constrains of the system. We will also losing generality, the gear ratio is assumed as 1.
realize a motion control and object handling with a good (ii) Servo-Brake Output Torque
performance comparable to an active type system.
We consider the case that the PRP is moving by forces
applied by the human operator or other system. φ̇w and
few denotes the angular velocity of and external force
applied to the wheel respectively. Ib is input current to
the brake and kb is the torque coefficient of the servo
brake. Let τw denote the resultant torque applied to the
wheel from the brake. Then τw will be:
a) for φ̇w = 0
τw = −kb Ib sgn(φ̇w ) (2)
Fig.2 . Hardware Design of PRP and Omni Wheel with MR Brake
b) for φ̇w = 0
III. PASSIVE TYPE ROBOT P ORTER PRP 
−few Rw |few |Rw ≤ kb Ib
We have developed a passive type Robot Porter system τw = (3)
−kb Ib sgn(few ) |few |Rw > kb Ib
based on the concept of the passive robotics and it is called
PRP-robot. PRP consists of three omni-directional wheels where, sgn(*) is the function to have sign of a param-
with servo-brakes to perform safety object transportation eter, and kb ≥ 0. Also as a brake, Ib ≥ 0.
task. The omni-directional wheel is equipped with several It is obvious that the characteristics of a brake-wheel
small rollers so that the wheel can generate driving force system are complicated compared with a motor-wheel sys-
along its rotational direction, but can move freely in its wheel tem. It is dependent on the wheel rotation. The sign of the
axis. Each omni-directional wheel is directly connected to a output torque of the wheel is decided by the direction of
servo-brake, and the three wheels are arranged to have 2π/3 the rotation of the wheel(Fig.3) and the magnitude of the
angle between each pair of wheel axes. A force/torque sensor
is installed on PRP for measuring the forces applied by the
human operator. It is necessary to note that the force sensor
is not indispensable to control of PRP if we do not need
to have precise dynamic characteristics, such as, impedance
characteristics of the system. Encoder is installed on each
wheel used for odometry. A computer system is installed for
controlling PRP and the system is powered by batteries.
The control performance of PRP system depends on the
characteristics of the servo brakes installed. In the first
prototype, we used MR Brake. The braking torque of MR
Brake is generated by chain mechanisms of iron powder from
free flow state, which are reacting to the applied magnetic
field. This provides a very reliable and linear braking torque,
and relatively small power consuming compared with motors. Fig.3 . Characteristic of Output Torque of Wheel with a Servo Brake

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 ThD4.1

torque is proportional to the input current of the brake while torque on each wheel will follow the same relationship with
the wheel is rotating. On the other hand, the torque will the input current on the servo brake.
be expressed as a non-linear function to the external force
B. Braking Torque and Statics of PRP
applied on the wheel in the case that the wheel does not
rotate. We call this the Singular Point of Braking Torque of We can express the relation between braking torque (tw =
T
a passive wheel. In object transportation, there are situations [τw1 , τw2 , τw3 ] ) generated by wheels and resultant braking
T
that a wheel does not rotate and they usually occur in a force and moment r F w = [r fx ,r fy ,r nz ] as follow
very short period. From the Eq.2, we can have the following tw = J T r F w (7)
condition between angular velocity of the wheel and braking
torque of the brake-wheel system. This static relationship is exactly the same to a system with
active actuators. Based on the wheel arrangement of PRP, J
τw φ̇w ≤ 0 (4) is full rank. However, we have to consider that the passive
This condition is the servo-brake control constraint to the actuators will apply different or opposing output torques
system and indicates that one cannot have arbitrary torque to the robot for different motion types. With the possible
from a servo-brake. We need to consider feasible braking torques for all motion types, the braking torque set V will
torque in the robot motion control based on this constraint. be the same as system with active actuators. This is shown
in Fig.5-(a), as a closed cube in the configuration space of
V. M OTION C ONTROL OF PRP
braking torque. 3
A. Kinematics and Motion Type of PRP
 
V = τvi ei  |τbi | ≤ τmax (8)
The kinematics
 relation between the motion vector of PRP,
i=1
q̇ = ẋ, ẏ, θ̇ , and angular velocity vector of wheels, Φ =
However, it does not mean that all torques in this set will
[φw1 , φw2 , φw3 ], can be express as: be feasible by setting proper control input since the robot
q̇ = J Φ (5) motion only belongs to a particular motion type in each
where J, moment. The servo-brake control constraint in Eq.4 should
⎡ ⎤ be included in the analysis. Here, we discuss the feasible
0 −R
√w
3
R
√w
3
J = ⎣ − 2R3w Rw Rw ⎦ (6) braking torque in each motion type. Uk denotes the set of
3 3
− R3Lw − R3Lw − R3Lw feasible braking torque when PRP robot is in k-th motion
type (k = 1, 2, · · · , 8), and A(Uk ) denotes the resultant force
Rw denotes the radius of the wheel and L denotes the and moment on the robot from the braking torque set Uk .
distance between center of the wheel and intersection point
3
 
of three axes of wheels in the horizontal plane. Robot Uk = τwi ei  |τwi | ≤ τmax , τwi φ̇wi ≤ 0} (9)
coordinates are shown in Fig.4. In addition, the Jacobian J i=1
3
is a full rank matrix. There is a unique mapping relationship
 
A(Uk ) = τwi v i  τwi ∈ Uk (10)
between the robot motion and wheel angular velocities. i=1
Since the brake torque of each wheel is dependent on the
where 
direction of the wheel rotation, we classify the motion of e1 e2 e3 = diag(1, 1, 1)
PRP into 8 different cases based on the signs of the angular  −1 
velocities of the three wheels (sgn(φ̇wi ), i ∈ 1, 2, 3). v1 v2 v3 = JT e1 e2 e3
k ∈ 1 ∼ 8 (P RP M otion Condition Case N umber)
Table.I M OTION T YPES AND C ONDITIONS OF PRP
Since PRP has eight different motion types, eight sets of
Sign of Angular Velocity of Wheel
Uk exist as the subset of set V and correspondingly, 8 eight
Wheel 1 + + + + - - - - A(Uk ) sets also exist. It is easy to know that V = k=1 Uk .
Wheel 2 + + - - + + - - But we want to note that Uj ∩Uk = 0(j, k = 1, 2, · · · , 8, j =
Wheel 3 + - + - + - + - k) and also A(Uk ) have the same propositions.
Motion Type No. 1 2 3 4 5 6 7 8 Fig.5-(b,c) show the set of Uk and A(Uk ) respectively
In each motion type, signs of the angular velocities of the when PRP is in Case 1. Uk is a subset of V just in one
three wheels will not change. Therefore, the feasible braking quadrant of the braking torque configuration space with six
plane constraints. The three constraint planes connected to
the origin of the coordinates are the braking torque con-
straints. The other three constraint planes are from maximum
torque limitation of each servo-brake. Since the feasible
resultant force and moment set A(Uk ) is the set projected
from Uk , each constraint surface of A(Uk ) set has the same
meaning. Based on the motion which belongs to one motion
type in Case 1∼8, feasible resultant general force (r F w )
and its corresponding braking torque tw could be determined
Fig.4 . Configuration and Robot Coordinates of PRP (Top View) uniquely.

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 ThD4.1

(a) (b) (c)

Fig.5 . (a) The Total Set of Braking Torque V (b) Wheel Braking Torque Set U1 (c) Feasible Resultant Force and Moment Set A(U1 )

C. Singularity of PRP Braking Torque desired braking torque tw d obtained from inverse dynamics
Since the omni-directional wheel incorporated in this sys- of the system. This is the same approach with an active type
tem rotates free on the axis direction of the wheel, v wi , the robot system.
velocity of wheel i will have two independent components,
velocity on driving direction, v wi drv , and velocity on passive
direction, v wi pas . It is well known that motion of PRP
will have an Instantaneous Center of Rotation, xICOR and
velocity of any point on the robot will be perpendicular
to the line connected to xICOR (Fig.6). According to the
discussion in the previous session, the singular point of
braking torque exists in the case that v wi drv = 0. In the
moment, v wi = v wi pas . Singular point of braking torque is
not unique in this kind of system. Here, let’s denote lsig wi
the set of points that the wheel i is braking singular where
instantaneous center of rotation xICOR is located on. In
PRP (Fig.6-(b)), lsig wi is a line parallel with the rotational
direction of the wheel i, and passing through the intersecting
point of the wheel and wheel axis.
From this geometric propriety, it is easy to identify the
Fig.7 . Derivation of Feasible Resultant Force and Moment for Control
singular point of the braking torque, and to check the
motion type of PRP when we control a PRP. Also from the On the other hand, there are cases in which the desired
geometric analysis, we can understand that there also exists force and moment r F w d is located outside the feasible set
points that two wheels are in singulars on braking torques A(Uk ), and cannot be generated by the passive actuators in
(Fig.6-(c)). In this moment, only braking torque of one wheel the current type of motion. One typical example is that a
can be controlled on the Eq.2. Other two wheels can not be passive type robot cannot generate force to accelerate the
control directly. Torques from those two wheels are governed object by itself whether it is moving or in halt mode.
by Eq.3. Above discussion on singularity on braking torque Because Uk is always in one of the quadrants of the brak-
can be applied to all other type passive robots with different ing torque space, it is not locally controllable around the ori-
wheel configurations. This makes the method described later gin of force and moment space of PRP. However, if there is a
be feasible to other passive mobile platforms. proper offset of the force/moment applied to the system, we
can have the local controllability around that force/moment.
D. Local Controllability and Motion Control of PRP Actually, this proper offset of the force/moment leading
During object transportation, the force and moment r F w d default control force/moment into inside of the set Uk is
which should be generated by the robot are determined a kind of resistant force/moment to the motion of passive
by the control law applied to the system such as motion robot PRP. Then, we can have the local controllability of
control for path tracking and obstacle collision avoidance, PRP where the motion of PRP is decelerated if there is no
impedance control, etc. For an active type robot, we just any other external force applied to the system.
simply command the motors of the robot to generate torques
for realizing this desired force and moment. However, to a E. Impedance-based Motion Control
passive type robot system, the feasible force and moment In this study, we consider that a human operator is always
is always dependent on its current motion. We need to pushing the object transportation system (Fig.8). This is
examine if the desired force and moment r F w d is in the important to us because this could not only let the object
feasible force and moment region in the current motion type, transportation task be achieved without losing speed, but also
which is determined by the sign of the angular velocities guarantee local controllability of PRP system during object
wheels(Fig.7). In the case that it is in the feasible region transportation, if we design the control algorithm properly.
A(Uk ), we can command the servo-brakes directly with In [12], we demonstrated the path following control of PRP

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 ThD4.1

(a) (b) (c)

Fig.6 . Instantaneous Center of Rotation of PRP and Singular Position: (b,c) Instantaneous Center of Rotation of PRP is Located on the Singular
Position: (a) Wheel 1 is singular on braking torque, (b) both Wheel 1 and Wheel 3 are singular on braking torques.

compliant coefficient matrix respectively. Then Eq.12 could

be rewritten as:
    
d qe 0 I qe
=
dt q̇ e −M −1 K −M −1 D q̇ e
 
0
+ F (14)
M −1
Fig.8 . (a)PRP with External Force and Moment during Object Transpi-
ration, (b)Relationship of r Fassist , r Fb and r Fw Here F is the resultant force of external force exclude the
part of assistant force for guarantee that the braking torques
which includes the motion perpendicular to the pushing are inside of the feasible force/moment set A(Uk ).
direction by the operator but that task did not need have
F = F ext + (F assist − F w ) (15)
detailed design of the interaction with human being.
Here, we are focusing on how to realize a desired apparent where F ext is the external force applied by the other one
dynamics of PRP. This is more challenging than tracking a or environment except the human operator. By realizing
path since it needs to have detailed design of the interaction the impedance characteristics on each PRP, the coopera-
with human being or environment. We consider impedance tive object handling strategy we proposed for active type
based apparent dynamics because it not only is useful on robots[1][3][10] can be applied to passive type system directly.
considering safety issue while the system has directly inter-
action with human being or environment, but also contribute VI. E XPERIMENTS
to realizing cooperative object handling with multiple PRPs. In this research, we demonstrated impedance-based motion
The dynamics of the whole system can be represented as: control in object transportation of PRP with human operator.
The human operator pushes the system along r x. Also it is
(M P RP + M Obj )r q̈ + D r q̇ + F kf (r q) = r F assist + r F w
pushed by other one in the perpendicular to the original path
(11)
during the transportation. In the experiment, the orientation
where, M P RP and M Obj denote robot’s and object’s mass,
of the object is controlled to keep a desired orientation (−90
D and F kf denote damping coefficient (including the vis-
degree) and a desired angular velocity of zero. Fig.9 ∼ Fig.11
cous friction of the motion) and coefficient of kinetic friction
show the results of the experiment. Fig.9 shows that PRP is
respectively. F assist is the force applied by the human
pushed by other one during object transportation.
operator and F w is the braking force and moment generated
by wheels of PRP. With a non-zero F assist which could
maintain the motion of PRP, F w will be a vector which is
not near the origin. It will be located inside of the feasible
force moment set A(Uk ), and the system will be locally
controllable. Then, we can control the motion of PRP under (a) (b) (c)
certain boundary of the braking force and moment (Fig.7) for
trajectory following and collision avoidance, as controlling
active type robots.
We consider realizing the following apparent dynamics to
the PRP system including the object on PRP.
(d) (e) (f)
Fig.9 . Experiment of Impedance-based Motion Control
M Δq̈ e + DΔq̇ e + KΔq e = F (12)
Δq e = qm − qd (13) Fig.10 shows path of PRP, and Fig.11 shows the force
applied to PRM in r y direction which is F e xt in Eq.15
where, q m and q d are desired position and orientation and the position and orientation of PRP. The experiment
of PRP without and with effect of extended force F re- results show that with only braking torque, PRP performs
spectively. M , D, K are apparent inertial, damping and good impedance-based motion characteristics (Fig.11-(b)) in

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 ThD4.1

(a) (b) (c)

Fig.10 . Impedance Based Motion Control: Path of PRP

5 0.3
applied force ( y direction ) [N] actual y position
desired y position
0 0.25

-5 y position [m] 0.2

force [N]

-10 0.15

-15 0.1 (d) (e) (f)

-20 0.05

-25 0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
time [s] time [s]

(a) rf (b) ry
y
4.5 -40
x position actual orientation
4 desired orientation

3.5 -60

3
(g) (h) (i)
orientaion [m]
position [m]

-80
2.5

2
-100
1.5

1 -120
0.5

0 -140
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
time [s] time [s]

(c) (d) r θ rx

Fig.11 . Experiment Result of Impedance Based Motion Control: Force in (j) (k) (l)
y direction and Trajectory of Position and Orientation Fig.12 . Orientation Control for Passing an Object through a Narrow Place:
Only the pulling force is applied to the center of the object-robot system.
the direction perpendicular to the moving direction. When PRPs rotate the object to an orientation cooperatively when they are passing
the human being pushes PRP (5.5sec ∼ 8.5sec), PRP is the narrow place, and more the object back to the initial orientation finally.
moving to compliant the push. After the human being stops of assistance force and geometric configuration of passive
his push (8.5sec), PRP is moving back to its original path. wheels. This is the problem of manipulability of passive type
In Fig.12, we demonstrate that two PRP robots control the systems with the push of the human operator. Some basic
object orientation cooperatively so that the object can pass investigations on this issue have been done to our prototype
through a narrow place. During the demonstration, a human PRP robot. Systematic analysis of manipulability and design
operator only pulls the PRP-object system by a wire and only of interaction force will be our future works.
pulling force is applied to the mass center of the object. The
moment for rotating the object is generated by braking force R EFERENCES
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