計測自動制御学会東北支部 第 202 回研究集会 (2002.7.2)
資料番号 202-4

Application of Reaction-less Motion to Teleoperation of a
Flexible Base Dual-Arm Manipulator○鷲野誠一郎 ∗ ，Michael Shoemaker∗∗ ，内山勝 ∗○ Seiichiro Washino∗ , Michael Shoemaker∗∗ , Masaru Uchiyama∗*東北大学, **Virginia Polytechnic Institute and State University*Tohoku University, **Virginia Polytechnic Institute and State University
キーワード : Flexible Structure mounted manipulator System, Reaction Null Space, reaction compensation
連絡先 : 〒 980-8579 仙台市青葉区荒巻字青葉 01 東北大学 大学院工学研究科 航空宇宙工学専攻
宇宙航空制御学講座宇宙機械学分野 (内山研究室)
鷲野誠一郎，Tel.: (022)217-6973， Fax.: (022)217-6971， E-mail: washino@space.mech.tohoku.ac.jp

1.

Introduction

ous Manipulator (SPDM) mounted on the Space
Station Remote Manipulator System (SSRMS)1) ,A robotic manipulator designed for space appliand the Japanese Experimental Module Small Fine
cations is strongly desired to be lightweight andArm attached to the Remote Manipulator Systemcapable of fast movement because of limitation of(JEMRMS)2) .

launch vehicle payload weight, restrictive launch
When we control this kind of system, problems
costs, and limited mission duration. In such sitassociated with ﬂexibility of the long-reach manipuations, a long-reach manipulator system, which
ulator emerge. Namely, when the rigid manipulaconsists of a series of small manipulators attachedtors move after being issued control commands, ato the end of a long manipulator, able to keep areaction force and moment are transferred to thelarge workspace and also do precise work, is probase of the long ﬂexible arm and subsequent vibraposed. We can regard the long manipulator as ation occurs. This vibration may bring performance
ﬂexible arm because of its low rigidity. Though its
penalties, so we must solve this problem of vibraaccuracy of work isn’t assured, it can keep a large
tion to operate it practically.
workspace because it has a large reachable range.
During practical operation, it is realistic that theOn the other hand, small manipulators attachedlong-reach manipulator is ﬁrst used to decide theto the end of the long manipulator don’t have ageneral position of the end-eﬀector, after which onlarge reachable range, but they can do more precise
ly the rigid manipulators work for the remainder ofand accurate work due to their high rigidity. Somethe task. In this case, the long-reach manipulator
notable examples are the Special Purpose Dexter-–1–

acts on a ﬂexible base on which the rigid manipulators are mounted. Therefore, when we focus ourattention on the rigid manipulators, we can regard
this vibration problem as similar to that of a Flexible Structure mounted manipulator System (FSMS). Many studies on controlling FSMS have beendone.
The experimental system designed in this lab
consists of a dual-arm manipulator, with each arm having two links and two joints and capable ofFig. 1Illustration of TREP

movement in a two-dimensional plane, mounted on
the end of a one link ﬂexible arm. This system,
named TREP, was previously used in both single
and dual arm conﬁgurations to study reaction-lessmotion and vibration suppression control with Reaction Null Space3, 4, 5, 7, 8) . Especially in the caseof a dual-arm conﬁguration, reaction compensation

A DC servomotor powers each joint on the rigid
manipulators. A strain gauge attached to the rootpart of the ﬂexible arm is used to measure the deﬂection of the end of the ﬂexible arm. A three axis
force/torque sensor is attached to the end of the
ﬂexible arm to sense the reaction force and moment

control was studied5, 8) . Finally, when applying

caused by the movement of the rigid manipulators.

these principles to the practical work of transport-

2.2

Control System of TREP

ing an object, the method using reaction-less moThe control system of TREP is shown in Figuretion and vibration suppression control was considered6) .
In this study, we consider the case where precise
work is accomplished by a human operator throughteleoperation. Speciﬁcally, one arm of the ﬂexible
structure mounted dual manipulator system is teleoperated freely, with reaction compensation control

2. The main computer used to control TREP isa 32 [bit] Dell Dimension 2100 PC (CPU: Celeron
800 [MHz]) which runs the real-time operating system VxWorks. The joint angles of the motors of
the rigid manipulators are sensed by the encoders.
The pulses from the encoders are sent to the motor

provided by the other arm automatically.

driver, which in turn uses the pulses both to feed-

2.

TREP Experimental Systemback the angular velocity and to send the value ofthe joint angle to the main computer. The signals
from the strain gauge on the force/torque sensor

2.1

TREP Experimental Robotand on the root part of the ﬂexible arm are sent

An illustration of TREP is shown in Figure 1.

to the main computer through the strain ampliﬁer

TREP consists of a dual-arm manipulator, withand the A/D converter. The main computer calcu-

each arm having two links and two joints and capa-

lates the values of each joint angle, and sends theble of movement in a two-dimensional plane, mount-values back to the motor driver through the D/A

ed vertically on the end of a one link ﬂexible arm.

converter.

–2–y (high stiffness direction)

O2 Workstation
CPU:MIPS R10000
175 [MHz]
OS:IRIXSpace mouse

0l2
Lower arm

m4

l2

Ethernet

I2
m3

lg2

0u2

Main computer
CPU:Celeron 800 [MHz]OS:VxWorks

xb

O

I1
m1

0l1
m0

Pulse counter

Flexible arm

D/A converter

m2

0u1
lg1

A/D converter

Upper armx(low stiffness direction)l1

Motor controller
EncoderFig. 3DC servo motor

Strain amplifier3.

Dynamics Model of TREP
and Equation of Reaction
Compensation Control

3.1

Dynamics Model of TREP

Strain gauge

Force/torque sensor

Fig. 2TREP

Control system of TREP

The dynamics of TREP can be considered as an

For us to send the command to one arm of TREP
by teleoperation, the command ﬁrst needs to be received by an input device and displayed to the operator in visual form. For this purpose, a Silicon
Graphics O2 workstation (CPU: MIPSR10000, 175[MHz]) running IRIX is used because of its powerful graphics rendering capability.

The object-oriented graphical toolkit Open Inventor is usedas the application programming interface. Motioncommands are input via a six DOF pointer called
Space Mouse manufactured by LogiTech. The refresh rate of the display and the sampling rate at
which input values are collected from Space Mouseare both once every 20 [ms]. At the same time,
the reference joint angle and the angular velocityof the upper arm are calculated, and then sent to
the main computer through socket connection.

Model of TREPFSMS that consists of a dual-arm rigid manipulator, with each arm having two links and two joints
and capable of movement in a two-dimensional plane,
mounted vertically on the end of a one link ﬂexible arm (Figure 3). Since both rigid manipulators
have the same layout and are attached verticallyatop one another on the base, we refer to them as
the upper arm and lower arm. The origin pointof the frame of this system is deﬁned as the point
where the end of the ﬂexible arm would be with zero vibration. We deﬁne the x-axis and y-axis of thecoordinate system as the low stiﬀness direction andhigh stiﬀness direction of the ﬂexible arm, respectively. Because of the feature of a parallel board
spring, we can assume that the orientation of theframe remains constant regardless of deﬂection of
the ﬂexible arm. The joint angles θu1 , θu2 , θl1 , θl2
are deﬁned as shown in Figure 3.

–3–

3.2

Thus, when F is zero, no reaction force is acting on

Equation of Reaction Compensation Control

the base. If we integrate (3) with respect to time,
the result is,

The equation of motion of the base can be expressed as follows in (1). Here, base deﬂection in

L = H bmu θ̇ u + H bml θ̇ l = const(4)

the x direction is xb , and base deﬂection in the y
If we make the assumption that there is no initial

direction is taken to be zero.

base movement, L is zero. Now, if we solve (4)H bmu (θu )θ̈u + H bml (θ l )θ̈ l + Cb (θ̇u , θu , θ̇l , θl )for θ̇l , considering that the lower arm compensates+Hb ẍb + kb xb = 0

the reaction force of the upper arm, the equation(1)

is expressed as,
where,
θ̇l = −H +bml H bmu θ̇ u2

θu ∈ Rθ l ∈ R2::

H bmu (θu ) ∈ R1×2:

(5)

joint angle of the upper arm
where H +bml denotes the right pseudo inverse of
joint angle of the lower arm
H bml . This solution expresses the angular velocity
inertia coupling matrixof the upper armof the lower arm which compensates the x compo-:

inertia coupling matrixof the lower arm

nent of the reaction force caused by the upper arm

Hb:elastic base inertia
coeﬃcientof TREP.

Cb (θ̇ u , θu , θ̇l , θ l ):non-linear Coriolis andcentrifugal terms

4.

kb

:

elastic base stiﬀness
coeﬃcientxb:

elastic base x direction
deﬂection

H bml (θl ) ∈ R

1×2

Inverse Kinematics and Jacobian of TREP

When we calculated the part of the command
dealing with the teleoperation of the upper arm,
we used the inverse kinematics and the inverse Ja-

Considering that C b is equivalent to Ḣ bmu θ̇u +

cobian matrix. Before explaining the method ofḢ bml θ̇l , we can divide (1) into θ u ， θl term andcontrol, we will outline these equations brieﬂy.

xb term.
H bmu (θu )θ̈u + H bml (θ l )θ̈ l + Ḣ bmu θ̇u + Ḣ bml θ̇l= −Hb ẍb − kb xb

4.1

Inverse Kinematics of TREP

The equation of kinematics of the upper arm can(2)

be easily derived from Figure 3, which is written as
From this, we can see that the left side of equation

follows,
(2) aﬀects the base deﬂection xb , so these terms
rau

can be regarded as the reaction force acting on the=base. Calling this reaction force F , the equation is=rewritten as,yau
l1 cos θu1 + l2 cos(θu1 + θu2 )l1 sin θu1 + l2 sin(θu1 + θu2 )(6)where r au is the position of the end point of the up-

F = H bmu (θu )θ̈u + H bml (θ l )θ̈ l+Ḣ bmu θ̇u + Ḣ bml θ̇ lxau

per arm, keeping in mind that the reference frame(3)of r au remains ﬁxed on the base. Applying the–4–cosine theorem to (6), the equation of inverse kine-

diﬀerentiated with respect to time, the equation ismatics is written as,

written as,


θu1

=

arctan

yau
xau



κ

∓ arctanθu2=ṙ au = J u θ̇u(9)where,

2 + l2 − l2x2 + yau
1
2
 au

κ
± arctan
2 − l2 − l2x2au + yau
12

where,



ṙ au

=

θ̇ u

=

T



ẋau

ẏau

θ̇u1

θ̇u2

T


2 + l 2 + l 2 )2 − 2((x2 + y 2 )2 + l 4 + l 4 )κ = (x2au + yau
au
au
1
2
1
2

Then, the Jacobian of the upper arm is written as,−l1 Su1 − l2 Su12 −l2 Su12
(10)
Ju =
l1 Cu1 + l2 Cu12
l2 Cu12

As can been seen above, κ is only deﬁned where,

where,

2 < 2
2
0 < x2au + yau= l1 + l2 . Now, we must choose fromthe above two equations the form of the inverse

Su1=

sin θu1

Su2=

sin θu2

Su12=

sin(θu1 + θu2 )Cu1=

cos θu1

Cu2=

cos θu2

Cu12=

cos(θu1 + θu2 )kinematics equation we want to use to express our
experimental conﬁguration. The equations used in
the experiment are,

θu1

=

arctan

yau
xau




=



(7)
2 + l2 − l2x2 + yau
1
2
 au

κ
− arctan(8)22xau + yau − l12 − l22+ arctanθu2κ

If we multiply both sides of (9) by J −1
u , the equation can be rewritten as,θ̇ u = J −1
u ṙ auExpressed in this manner, negative values of θu2
decide the posture of the arm. With these equa-

where,

tions, there is a one to one correspondence of the

J −1
u =

end point position of the upper arm and the joint

1
l1 l2 Su2(11)

l2 Cu12l2 Su12−l1 Cu1 − l2 Cu12−l1 Su1 − l2 Su12

With this equation, there is a one to one correangles, and we can calculate the joint angles fromspondence of the upper arm end point velocity tothe position. It can also be seen that the singular
the joint angular velocity, and thus the joint angupoint of this equation is the origin, meaning whenlar velocity can be calculated from the end point
the upper arm is positioned so that the end point
velocity. It can also be realized that the singularis located at the origin, the solution is undeﬁned,
positions are points such that,and the one joint angle can’t be decided by the oneθu2 = nπposition.

4.2(n = 0, 1, 2, . . .)These positions correspond to the manipulator ei-

Jacobian of TREP

ther being fully extended or folded in on itself. As

As explained previously, rau is the position of thethe manipulator approaches these conditions, theend point of the upper arm, where the reference

closer to inﬁnity the value of the joint angular ve-

frame of rau remains ﬁxed on the base. If (6) islocity becomes.

–5–


(12)

Space mouserau _ ref

Numerical
Integration

rau _ ref Inverse
Kinematics

θ u _ refθu = J u-1rauθ u _ cur
-

+

θ


u _ ref+θl = -Η bml
Η bmuθuθ
l _ ref

Numerical
IntegrationKpθ l _ ref
+

FSMS

Kp

-

θ l _ curFig. 4Block diagram of the control
Fig. 5

CG display of upper arm of TREP

5.

Experiment of Reaction Comangular velocity of the upper arm θ̇ u ref and thepensation Control by Telecurrent joint angle of the lower arm θl cur to calcuoperationlate the reference joint angular velocity of the lower

5.1

Method of Control and Block Diagram

cally integrated with respect to time to yield theIn this experiment, the upper arm is moved usingreference joint angle of the lower arm θl ref . Afterthe velocity command generated by Space Mouse,

this, the upper and lower arm move according toand the lower arm is moved using (5). The block

their respective reference angles with proportional

diagram for this control is shown in Figure 4, wherecontrol.

arm θ̇ l ref . This angular velocity is then numeri-

By using the above described control method, it

Kp expresses a proportional gain.
The following calculations are done in the O2

is possible for the operator to control the upper

workstation. First, the reference velocity of thearm at any desired reference velocity using Space

upper arm ṙau ref is taken from Space Mouse andMouse, causing the lower arm to compensate thenumerically integrated with respect to time to yieldreaction force.

the reference position of the upper arm, rau ref ,5.2

which is used for the following: the reference joint

Visual Display of Command Input

angle of the upper arm θu ref is calculated using
When movement commands are issued to TREP

the inverse kinematics equations, and the reference
joint angular velocity of the upper arm θ̇ u ref iscalculated with (11) using both the reference velocity of the upper arm ṙ au ref and the reference

via the Space Mouse, the operator is working froma location removed from that of the actual TREP
unit. Therefore, the O2 workstation is used to draw
a CG display of the upper arm of TREP as a visual

joint angle of the upper arm θu ref .
The next part of the control scheme is calculated
in the main computer. After the reference joint an-cue to the operator. The CG display is shown in
Figure 5.
The teleoperating system of TREP is shown in

gle of the upper arm θu ref is calculated, it is usedaccording to equation (5) with the reference joint

Figure 6. The refresh rate of the display and thesampling rate at which input values are collected–6–per arm mustn’t be allowed to approach this limit

CG Display

TREP

too closely, because as the arm approaches its fully
extended posture, the value of the reference joint
angular velocity approaches inﬁnity. Thus the reau_cur

u_ref

l_cur

l_ref

Calculation

son for the boundaries as shown on the CG display.
u_refDrawing CG

u_ref

Calculation

l_ref
u_ref

256 [Hz]
Main Computer
OS : VxWorks,

rau_ref

5.3

Experimental Conditions

u_ref

50 [Hz]In the ﬁrst experiment, the upper arm is moved
Space Mouseas described previously by the velocity command

O2 Workstation
OS : IRIX

created by Space Mouse. The experimental conditions are as follows,

Fig. 6

Teleoperating system diagram of TREP
• Initial position

from Space Mouse are both once every 20 [ms], inUpper arm:θu =other words, at a frequency of 50 [Hz]. After that,

Lower arm:θl =the reference joint angle and the angular velocity




180◦
0◦

−90◦
T

T

90◦

• Time duration = 5.0 [s]of the upper arm are calculated as described previously, and then sent to the main computer through• Sampling frequency = 256 [Hz]socket connection. The reference joint angle is also

• Proportional gain Kp = 100 [s−1 ]used to render the upper arm on the display, mak• Lowpass ﬁlter cutoﬀ = 2.0 [Hz]ing it possible for us to show the reference velocity
to the operator at a rapid rate.

Despite the high sampling frequency of 256 [Hz],The two half circles on the display represent thethe movement commands generated by the opera-boundaries in which the upper arm can move. We

tor are collected and sent to the main computer atdetermined this range as,a rate of only once per 20 [ms]. This fact caused

22x2 + y 2 <= 360, and (x + 10) + y >= 95, and y >=0

movement commands that were too jagged in appearance, so our solution was to use a single-orderThis is intentionally set slightly smaller than theinterpolation in the main computer to approximate

actual mechanical range of the upper arm so as not

position values.

to let the upper arm approach the singular points

As a comparative experiment, we repeated thethat we have explained before. We didn’t consider

ﬁrst experiment but with no reaction compensa-

the singular point of the equations of the inversetion control supplied by the lower arm. The samekinematics because we assumed that there is little

upper arm movement commands from the ﬁrst ex-

chance that the operator would generate the com-periment are reused so that the only diﬀerence be-mands necessary to approach the speciﬁc singulartween the two trials is the presence of lower arm

point. However, the singular points of (11), espe-

movement.

cially the case where the arm is fully extended, caneasily be caused by the operator. Hence, the up-

–7–

0.2

0.2

Start Pointya [m]

ya [m]0.1
0-0.1

changes direction along the x-axis rapidly as seen

Start Point

0.1

in part (a) of Figure 9. It follows intuitively that an

0

acceleration of the manipulator in the low-stiﬀness-0.1

direction in the absence of any reaction compensation will cause a force and a subsequent vibration of

-0.2

-0.2
-0.2

-0.1

0

xa [m]

0.1(a) Upper arm
Fig. 7

0.2

-0.2

-0.1

0

0.1

0.2

xa [m]

the base. Therefore, the fact that there is little basedeﬂection in the presence of high acceleration of the(b) Lower arm

manipulator as seen in the ﬁrst experiment shows

The movement of the both arms

that the reaction compensation control provided by

5.4(5) is eﬀective. It should be noted, however, that

Experimental Results

slight vibration of the base can still be seen even

The movement of both arms in the experimentwhen reaction compensation is used. The cause ofwith reaction compensation control are shown inthis vibration may be the big and sudden change

Figure 7. Plots of the data collected from both ex-of the reference joint angular velocity of the up-

periments are shown in Figure 8. Parts (a) andper arm when its value is also big. The command(b) of the ﬁgure show the upper arm joint angles,

created by teleoperation may tend to be like this.

θu1 and θu2 , respectively. Parts (c) and (d) show

When the value of the reference joint angular ve-

the lower arm joint angles, θl1 and θl2 , respective-

locity is big, the reference joint angle calculated asly. Part (e) shows the x and y components of thethe command is rough, and more when the value

reaction force, Fx and Fy , caused by manipulatorof the reference joint angular velocity changes sud-

movement. Finally, part (f) shows the deﬂection of

denly, the accuracy of realizing (5) may be going

the ﬂexible arm. The results from the experimentdown. In this experiment, we can see the suddenwith no reaction compensation control are likewisechange of the reference joint angular velocity atshown in Figure 9.

around time from 1.5 to 2.0 [s] in both joint of the

5.5

upper arm, and this seems to make the vibration.

Interpretation of Results

Second, we will consider the results in terms ofFirst, we will consider the results in terms of thethe experimental teleoperation of TREP. In our ex-

performance of the experimental reaction compen-

periments, the operator relied on a CG representa-sation control of TREP. In the case where reaction

tion of the movement commands to control TREP.

compensation control is used, the force in the low

We found that it was eﬀective to show the mov-rigidity direction, Fx , remains low, and the maxi-

able range in the display, including the consider-mum deﬂection of the base does not exceed 1 [m-ation of the singular points of the upper arm, as

m]. Conversely, in the case where there is no reac-

a way to let the operator choose appropriate com-tion compensation, the same upper arm movement

mands. While still developing the system, we foundcauses greater Fx and a greater maximum deﬂec-

that it was a very diﬃcult task to control the up-tion of almost 3 [mm]. This is most pronounced at

per arm by simply looking at TREP without thearound time 2.0 [s], when the upper arm velocity

–8–

-60

0

2

1

3

4

-150

5

1

2

3

4

5

1

2

3

4

-150

5

1

0

2

(b) θu2(a) θu1

120

3

120

30

90

Reference
Current
l2 [deg]

l1 [deg]

l2 [deg]905(b) θu2
Reference
Current

120

4Time [s]Time [s]

60

0

90

Reference
Current

30

0

2

1

3

4

60

5

0

1

2

(c) θl1

4-305(d) θl2
Fx
Fy

Deflection [mm]

-1

3

4

Time [s](e) Reaction force
Fig. 8

5

3

1

0

-5

2

2

4

1

2

0

1

3

2

4

5

4

5

5

0

0

-5
1

2

3

4

5

0

Time [s]Time [s](e) Reaction force(f) Base deﬂection (x)Results of experiment with compensation

4(d) θl2
Fx
Fy

0

5

3

Time [s]

-1
0

60

5

(c) θl1

5

0

1

1

0

Time [s]Force [N]1

3Time [s]Time [s]Deflection [mm]0Force [N]0Reference
Current

150

-90

-120

Time [s](a) θu1

0

150

90
0

Time [s]180Reference
Current

120

-120

120

-60

Reference
Current
u2 [deg]-90

u1 [deg]

15090

l1 [deg]

180

Reference
Current

Reference
Current
u2 [deg]

u1 [deg]

180

Fig. 91

2

3Time [s](f) Base deﬂection (x)Results of experiment with no compensa-tion
CG display. It was especially easy to inadvertently

This is where a CG display of the movable range

approach the outer boundary, represented by theof the manipulator becomes a necessity. Also, inlarge half-circle, making manipulation tasks very

the CG display used in our experiments, the mov-diﬃcult.

able range was shown for the upper arm, but noThere are additional factors that should be con-consideration was given to displaying the movablesidered when implementing a CG display as we did.

range of the lower arm, which compensated the re-In cases where a robot manipulator has a relative-

action force. As a result, even if the upper arm wasly simple conﬁguration of links and joints–as does

operated within its limits, sometimes the lower ar-

TREP, with two links and two joints per arm–it

m would move in an undesirable fashion, causing

is somewhat easy for the operator to imagine themore vibration than it was supposed to compen-

singular points and the limits on movable range

sate. This highlights the additional need, in casesof the manipulator being controlled. But in cas-similar to our experimental system, for a display

es where the manipulator has a complicated struc-that incorporates the movable ranges of both theture, it might be too diﬃcult for the operator to do

directly and indirectly controlled arms teleoperated

this mentally while performing teleoperation tasks.

manually by the human operator.

–9–

6.

Conclusion

We have considered the case where precise work
is accomplished by a human operator through teleoperation. With the experimental system TREP,
we teleoperated one arm of the ﬂexible structure
mounted dual manipulator system freely with reaction compensation control provided by the otherarm automatically.
From the results, in terms of the performance ofthe experimental reaction compensation control of
TREP, we can recognize the method of control is effective. But, because of the inability of the system
to realize control equation (5) under certain conditions, small vibration occured even with the lower
arm compensating. When we consider the resultsin terms of the experimental teleoperation, in cases
similar to our experimental system, we should needa display that incorporates the movable ranges ofboth the directly and indirectly controlled arms
teleoperated manually by the human operator.

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Konno, M. Uchiyama: Experiments on Reaction
Null-Space Based Decoupled Control of a Flexible
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