2011 IEEE International Conference on Robotics and Automation
Shanghai International Conference Center
May 9-13, 2011, Shanghai, China

A Torque Cancelling System Using the Parallel Solution Scheme
Daigoro Isobe, Yasumasa Matsui, and Kensuke Kondo

Abstract— A new torque cancelling system (TCS) thatstabilizes mechanical sway in quick-motion robots is discussed
in this paper. It cancels the reaction moment generated by the
motion of an object by considering the precise dynamics of theobject and the body of the robot itself. The reaction moment
can be obtained accurately using the parallel solution schemeof inverse dynamics, which handles the dynamics of complex
robotic architectures by modeling them with finite elements.
Once the reaction moment is known, it can be cancelled byapplying an anti-torque to a torque generating device. In thispaper, the general concepts of the TCS and the parallel
solution scheme are first described. Then, some examples of
torque cancelling due to accurate calculations of dynamics aredemonstrated, by showing the experimental results carried out
on a prototype TCS system. The objects used in theexperiments include rigid and flexible, outboard and inboard
links, where difficult assumptions are normally required toconsider the accurate dynamics.

A

I. INTRODUCTION

new torque cancelling system that stabilizes
mechanical sway in quick-motion robots is discussed inthis paper. Mechanical sway generated by reaction
moments occurring in the motion, for example, of spacestations is a big issue that can not be neglected. Motion
controls using the conservative law of angular momentum
are applied by driving control momentum gyros (CMG) [1]or reaction wheels [2]. Applications of the gyros and thewheels are generally limited to slow momentum transitions,relying upon high-gain feedback control. Although the same
strategy can be used in the motion control of robots when the
momentum transitions are not so fast, the dynamics of therobots should be considered to effectively control quicker
motions. The torque cancelling system (TCS) discussed inthis paper cancels the reaction moment generated by the
motion of an object by considering the precise dynamics ofthe object and the body of the robot itself. The reaction
moment can be obtained accurately using a newly proposedsolution scheme of inverse dynamics, which handles the
dynamics of complex robotic architectures by modeling

Manuscript received August 25, 2010.
D. Isobe is an associate professor of the Department of Engineering
Mechanics and Energy, University of Tsukuba, Tsukuba-shi, Ibaraki
305-8573, Japan. e-mail: isobe@kz.tsukuba.ac.jp.
Y. Matsui is a master degree graduate student in the Department ofEngineering Mechanics and Energy, University of Tsukuba, Tsukuba-shi,
Ibaraki 305-8573, Japan. e-mail: e0712350@edu.esys.tsukuba.ac.jp.
K. Kondo is also a master degree graduate student in the Department ofEngineering Mechanics and Energy, University of Tsukuba, Tsukuba-shi,
Ibaraki 305-8573, Japan. e-mail: e0611350@edu.esys.tsukuba.ac.jp.

978-1-61284-380-3/11/$26.00 ©2011 IEEE1064

them with finite elements. Once the reaction moment is
known, it can be cancelled by applying an anti-torque to a
torque generating device.
The newly proposed solution scheme of inverse dynamics
mentioned above is called the parallel solution scheme [3],and it was developed on the basis of a finite element
approach. By taking advantage of the natural characteristicsof the finite element method, i.e., the capability of
expressing the behaviour of each discrete element as well asthat of the entire continuous system, local information suchas nodal forces and displacements can be calculated inparallel. In this scheme, the analyzed model is evaluated inabsolute Cartesian coordinates with the equation of motion
expressed in the dimension of force. The nodal forces arecalculated incrementally in matrix form, which does not
require any revision of the overall frame, and the variables
forming the frame can be revised by simply changing the
input data in the case of a physical change in the hardware
system. The calculated nodal forces are then converted into
joint torques using a matrix-form equation divided into
terms of force, the transformation between the coordinates,
and length. In contrast to the conventional and revised
schemes that use dynamic equations [4]-[6], the structure of
the algorithm makes it seamless in its application to different
types of link system under various boundary conditions suchas open- or closed-loop link systems. It can also consider theelasticity of constituted links or passive joints by onlychanging the input numerical model, without the need torevise any part of the scheme. Its validity in various
feedforward control experiments of various kinds of linksystems has been verified [7], [8].
In this paper, the general concept of the TCS is firstdescribed. Then, an outline of the inverse dynamics
calculation scheme for handling accurate dynamics is shown,along with some examples of torque cancelling experiments
carried out on a prototype TCS system. The objects used in
the experiments include rigid and flexible, outboard andinboard links, where difficult assumptions are normallyrequired to consider the accurate dynamics.
II. TORQUE CANCELLING SYSTEM
A robot swinging around an arm in a quick motion, forexample, normally needs a counterbalance motion to avoid
any mechanical sway. The TCS proposed in this paper is asystem that generates anti-torque to suppress the mechanicalsway, by computing accurate inverse dynamics of the arm
motion with an accurate consideration of the dynamics of theoverall body is at the offset position from the rotating axis.
To cancel these moments, three TCSs, each set on eachdimensional axis, should be placed as shown in Fig. 2.
However, all of the TCSs do not have to be placed at one
place or placed exactly on the rotating axis, to maintain thefunction. Actually, a TCS can be mounted, literally,
anywhere in the body. All a TCS has to do to suppress amechanical sway is to cancel the moment generated at the
precise location of the TCS.

Rotor

Motor

TorqueCancelling moment

Reaction moment
Cancelling torqueIII. PARALLEL SOLUTION SCHEME OF INVERSE DYNAMICS
TCS
Fig. 1. General concept of uniaxial TCS.

TorqueReaction moment

Cancelling moment

Cancelling torque
TCS
Fig. 2. General concept of triaxial TCS.

robot itself. The accurate consideration of the dynamics iscarried out by implementing the parallel solution scheme
which is to be explained briefly in the next section.
A simple explanation of the TCS is drawn in Fig. 1 for theuniaxial case. Suppose a rotor is rotated by a motor with a
specific torque. A reaction moment will occur around therotating axis which will make the whole body twist aroundthe axis. To avoid the reaction moment from making thetwist, a cancelling moment can be generated by anothermotor placed on the axis by supplying a specific, cancelling
torque. The same concept is used in twin-rotator helicopters,
where two rotators rotate in opposite directions to cancel thetwist motion. However, the difference between thetwin-rotator helicopters and the proposed TCS is that the
TCS is driven using accurate torques by considering the
dynamics related to the motion and the overall architecture,
while the twin-rotators only rotate in opposite directions
with the same rotational speed.
A general concept of triaxial TCS is drawn in Fig. 2. If an
object in motion is considered as an outboard rotor (or acantilever beam as shown in the figure), the reaction
moments acting on the body become not as simple as thosein the uniaxial case. The moments will act aroundthree-dimensional axes because the center of gravity of the1065

In this section, an outline of the parallel solution schemeof inverse dynamics for considering accurate dynamics ofthe system is described. In the parallel solution scheme, alink system consisting of motor joints and links is modeled
and subdivided using finite elements. A link is substituted
with a single Bernoulli-Euler beam element, assuming aconsistent mass distribution along the link [8]. The
consistent mass matrix of a beam element is formulated in
the same manner as the displacement function and does not
require an expression of the center of gravity. This type ofmodeling has the merits of reducing computational time
without lowering accuracy, particularly against elastic
deformations. It requires only one-element subdivision per
member for cases of infinitesimal deformation because thedeformation of the element is defined using a high-orderdisplacement function.
Figure 3 shows the nodal forces based on global
coordinates acting on the i th link in a three-dimensional
open-loop n-link system with a consistent mass distribution.
The joint torque τix required around the x-elemental axis on
the i th link is determined by adding the i+1 th joint torqueτ(i+1)x to the sum of the moments of inertia acting on this link
and is expressed by the nodal forces based on elemental (orlink) coordinates asúix = li (

n
X

j=i+1

Pn

Fj )y + Fiûx + ú(i+1)x ; (i; j = 1 ò n);(1)

j=i+1 FjZ

FiûZ

Pn

j=i+1 FjYFiûY
Link iFiûX

Link i + 1
Pn
j=i+1 FjX

y

z

Z
Y

x
Elemental coordinates

X
Global coordinates

Fig. 3. Nodal forces acting on i th link in n-link system with consistentmass distribution.

P
where l is the link length and F is the nodal force. ( F )indicates the resultant force, the subscripts i, j the link
number, x, y, z the elemental coordinate components, X, Y, Z
the global coordinate components, and φ the angularcomponents. The right hand side of (1) becomes different in
the scheme with a lumped mass distributed at the center ofgravity [3]. By considering the other components around they- and z-axes and arranging them into global coordinates (X,Y, Z) in matrix form, the joint torque vector is expressed as
n

n

n

n

where {P } is a 6n×1 vector related to the nodal force and isdefined as
9
8 P
n
>
>
>
j=i+1 FjX >
>
>
>
>
Pn
>
>
>
>
F
>
>
jY
j=i+1
>
> P
>
>
n
=
<
F
jZ
j=i+1
:
where fPi g =
>
>
FiûX
>
>
>
>
>
>
>
>
>
>
FiûY
>
>
>
>
>
>
;
:
F
iûZ

(3)

[Tn] is a 6n×6n transformation matrix and is defined aswhere [hn] is a correction matrix between the x-y and z-x
coordinate systems, which simply inverts the signs of thecomponents in the y-axis direction. [TnGE] is atransformation matrix between the global and elemental
coordinates and is expressed as
6
6
6
6
h
i 6
6
n
TGE = 6
6
6
6
6
4

3

T1
T2

0
T3
Å

0

where
h

Ti

i

=

"

Å

Ai

0

0

Ai

2

cosûiXx6= 4 cosûiXy
cosûiXzÅ#cosûiY xcosûiY ycosûiY z

3cosûiZx
7cosûiZy 5 ;cosûiZz(6b)where φiXx, for example, represents the rotational angle
between the X (global) and x (elemental) coordinates.
[Ln] in (2) is a 3n×6n matrix related to link length and isexpressed as(7)[Ln ] = [TÉn ][Én ];where [TnΛ] is a transformation matrix between eachelemental coordinate and is expressed as
2

6
6
6
6
h
i 6
6
n
TÉ = 6
6
6
6
6
4

T11

Tn

7
7
7
7
7
7
7;
7
7
7
7
5

(5)

1066

T22

T23

3

7
Å Å Å T2n 77Å Å Å T3n 777
Å Å Å Å 7:7
Å Å Å 7
7
7
Å Å 5
Tnn
Å Å Å T1n

0

(8)

(9)

[Tij ] = [Ai ][Aj ]T :[Λn] is a matrix expressed as
2

6
6
6
6
h
i 6
6
Én = 6
6
6
6
6
4
h

and

T13

[Tij](i,j=1~n) is expressed using matrix [Ai] shown in (6b) as

where

(6a)

T12

T33

(4)

n
[T n ] = [hn ][TGE
];

2

Ai

i

(2)

n

fú g = [L ][T ]fP g;

8
9
P1 >
>
>
>
>
>
>
>
>
>
>
< P2 >=
fP n g =
Å ;
>
>
>
>
>
> Å >
>
>
>
>
>
:
;
Pn

h

Éi

i

3

É1
É2

0
É3
Å

0

2

0
6
= 4 li
0

li
0
0

Å

Å

Én

7
7
7
7
7
7
7;
7
7
7
7
5

3
0 1 0 07
0 0 1 0 5:
0 0 0 1

(10)

(11)

Consequently, joint torques can be obtained, as shown in(2), first by converting the nodal forces in global coordinates
to elemental coordinates and next by multiplying the matrix
regarding link lengths. The nodal forces are calculated bysupplying accurate target motions that compensate for theinertial forces acting at the links. Therefore, a kinematics
solution scheme using the finite element approach was also
developed to handle the elasticity of the links. See thereferences [3], [7] and [8] for more details.

Z

X
Fig. 6. Finite element subdivision of the setup.

7(a). Reaction moments generated at the exact location of
the TCS (lower node of element No. 1 in Fig. 6) arecalculated as shown in Fig. 7(b). To cancel these moments,
the torque shown in Fig. 7(c) should be supplied to the TCS.
Note that the cancelling torques the TCS has to generate area perfect duplicate of the reaction moments (see Fig. 1).
Figure 8 shows the rotational angle of the frame bodydetected at the encoder attached on the pin joint when the
Torque [N*m]Fig. 4. Experimental setup of a rotor mechanism with uniaxial TCS
mounted on the rotating axis.

τx
τy
τz

0.02

0

Fig. 5. Layout of a rotor and a TCS.
-0.02

IV. TORQUE CANCELLING EXPERIMENTS

0

1067

1

1.5
2
Time [s]Moment [N*m](a) Motion torque for driving the rotor.

Mx
My
Mz

0.02

0

-0.02
0

1.5
2
Time [s](b) Reaction moment generated at the exact location of TCS.
Torque [N*m]

A. Rigid Rotor - Uniaxial TCS System
A simple rotor-TCS experimental setup, as shown in Figs.
4 and 5, was made to carry out a numerical estimation andsome experiments for verification. In this setup, a rotor isdriven by a gearless motor, and a prototype TCS is placed
underneath, exactly on the rotating axis of the rotor. The
frame body is connected to the ground with a rotation-free
pin joint, where an encoder attached to a nonacutated motor
is used to detect the rotational angle of the frame body. The
rotor is made of aluminum, and the frame body is made ofstainless steel, both with enough sectional area such that thewhole architecture can be assumed to be rigid without any
elastic deformation.
Figure 6 shows the finite element subdivision of theexperimental setup, with each connection to connection
modeled with a single Bernoulli-Euler beam element. The
whole system is modeled with a total of 21 elements and 22
nodes. The capabilities of modeling this kind of complex
architecture and computing the dynamics are the mainfeatures of the parallel solution scheme.
A rotational motion of 4π rad in 2 s is given to the rotor.
The torque required for driving the rotor, which is actually
computed at the upper node of element No. 3 in Fig. 6, isobtained by the parallel solution scheme as shown in Fig.

0.5

0.5

1

τz

0.02

0

-0.02
0

0.5

1

1.5
2
Time [s](c) Input torque for TCS.
Fig. 7. Input torques and generated moments in the system.

Angle [rad]with TCS
without TCS

1

0.5
0

-0.5
0

0.5

1

1.5
2
Time [s]Fig. 8. Rotational angle of the frame body around the pin joint during the
motion.

Fig. 10. Layout of a rotor and a TCS.

motion torque is actually supplied to the rotor. It can be
confirmed from the figure that the frame body is unstable
during the motion when the TCS is not activated, whereas
the rotation around the pin joint stops perfectly when the
TCS is activated, which means that the reaction moments
generated around the rotating axis are cancelled perfectly bysupplying the accurately computed cancelling torques to theTCS.

XTorque [N*m]Fig. 11. Finite element subdivision of the setup.
τx
τy
τz

0.02

0

-0.02
0

0.5

1

1.5
2
Time [s](a) Motion torque for driving the rotor.
Moment [N*m]B. Rigid Rotor in Offset Position - Uniaxial TCS System
Next, an experiment was carried out on a different
configuration of the frame body, as shown in Figs. 9 and 10.
In this case, a TCS is mounted in the offset position of therotating axis to see if such a condition affects the capabilityof the system. The whole system is modeled with a total of
25 elements and 26 nodes as shown in Fig. 11. The samerotational motion is given to the rotor as in the previous
experiment, and, consequently, there is no difference
between Fig. 7(a) and Fig. 12(a). The motion torques fordriving the rotor are the values calculated at the upper nodeof element No. 3, and the reaction moments are the values
calculated at the upper node of element No. 25. The reaction
moment and the input torque for the TCS around the z axis in
Figs. 12(b) and 12(c) perfectly agree with those in Figs. 7(b)and 7(c). On the other hand, My in Fig. 12(b) is constantly
shifted to the positive direction because a moment due togravity subjected along the offset length is now generated.
The moment generated by gravity, however, is supported at
the pin joint and is not meant to be suppressed in this

Z

0.5
0.4
Mx
My
Mz

0.3
0.2
0.1
0
0

0.5

1

1.5
2
Time [s]Torque [N*m](b) Reaction moment generated at the exact location of TCS.
0.5
τz

0.4
0.3
0.2
0.1
0
0

Fig. 9. Experimental setup of a rotor mechanism with uniaxial TCS
mounted in the offset position of the rotating axis.

1068

0.5

1

1.5
2
Time [s](c) Input torque for TCS.
Fig. 12. Input torques and generated moments in the system.

Angle [rad]with TCS
without TCS

0

-0.1

Fig. 16. Finite element subdivision of the setup.

0.5

1

1.5
2
Time [s]Fig. 13. Rotational angle of the frame body around the pin joint during the
motion when a TCS is mounted in the offset position.

experiment.
It can be confirmed from Fig. 13 that also in this case, themechanical sway around the pin joint is successfully
suppressed to a minimal amount by activating the TCS
mounted in the offset position.

C. Flexible Link - Uniaxial TCS System
Lastly, a simple flexible link with TCS experimental setup,
as shown in Figs. 14 and 15, was made to carry out anumerical estimation and some experiments for verification.
In this setup, a flexible link is driven by a gearless motor,
and a prototype TCS is placed at an offset position from therotating axis of the active joint. The link system is connectedto the ground with a rotation-free passive joint, where an
encoder attached to a nonacutated motor is used to detect therotational angle of the overall body. The flexible link ismade of poly-carbonate, where the elastic deformation
occurred in the link should not be neglected in the inverse
dynamics calculation. The experimental setup is subdivided,with each connection to connection by a single

Bernoulli-Euler beam element. The whole system ismodeled with a total of 4 elements and 5 nodes as shown in
Fig. 16. The capabilities of modeling this kind ofarchitecture with flexible link and computing the dynamics
are, again, the main features of the parallel solution scheme.
A rotational motion of π rad in 2 s is given to the active
link. The torque required for the active joint is obtained by
Torque [N*m]

0

0.01

0

-0.01
0

0.5

1

1.5

2

Time [s](a) Motion torque for driving the flexible link.

Moment [N*m]

-0.2

0.04
0.02
0

-0.02
-0.04
0

0.5

1

1.5

2

Time [s]Torque [N*m](b) Reaction moment generated at the exact location of TCS.
Fig. 14. Experimental setup with uniaxial TCS.
Rigid link
Passive joint
TCS
Flexible link

0.04

flexible model
rigid model

0.02
0

-0.02

Active joint-0.04

Encoder

0

0.5

1

1.5

2

Time [s](c) Input torque for TCS.
Fig. 17. Input torques and generated moments in the system.

Fig. 15. Configuration of the setup and location of a TCS.

1069

Angle [rad]ACKNOWLEDGMENT

with TCS (flexible)
with TCS (rigid)
without TCS

0.1

The author wishes to acknowledge the contributions of
the following former and present graduate students of hislaboratory: Takeo Ueda, Daisaku Imaizumi, Youichi
Chikugo, Shunsuke Sato, Atsushi Yagi, Akihiro Kato, Koji
Yamanaka, Hiromasa Ueda, Yuto Kitamura, and Junya
Hayakawa.

0-0.1

REFERENCES
0

0.5

1

1.5
2
Time [s][1]Fig. 18. Rotational angle of the link system around the passive joint during
the motion.

the parallel solution scheme as shown in Fig. 17(a). Reaction
moment generated at the exact location of the TCS iscalculated as shown in Fig. 17(b). To cancel this moment,
the torque shown in Fig. 17(c) should be supplied to the TCS.
Here again, note that the cancelling torque the TCS has togenerate is a perfect duplicate of the reaction moment (seeFig. 1). A torque curve for the TCS when the active link isassumed as a rigid member, without any elastic deformation,is also shown in the figure. This torque curve is also supplied
to the TCS to see how the accurate consideration of
dynamics is important in the torque cancelling.
Figure 18 shows the rotational angle of the link system
detected at the encoder attached on the passive joint, whenthe motion torque is actually supplied to the active joint. It
can be confirmed from the figure that the link system isunstable during the motion when the TCS is not activated,
whereas the rotation around the passive joint stops perfectly
when the TCS is activated (by the torque obtained byassuming the elasticity of the flexible link), which meansthat the vibrating reaction moment generated around therotating axis are cancelled perfectly by supplying theaccurately computed cancelling torque to the TCS. Also, it
can be confirmed that the torque curve obtained by notassuming the elasticity of the flexible link, is not sufficientfor the TCS to suppress the reaction moment generated
around the passive joint.
V. CONCLUSION
The torque cancelling system proposed and developed inthis paper succeeded to suppress a mechanical swaygenerated by the motions of rigid or flexible objects. The
suppression was realized by considering the accurate
dynamics of the objects and the main body itself using a
newly developed calculation scheme of inverse dynamics.
As a consequence, innovation of the finite element approachto the inverse dynamics calculation enables us to consider
more precise dynamics of robots, which leads to more
precise feedforward control and cancellation of mechanicalsway during quick motions. Further simulations andexperiments on triaxial cases are scheduled. Furthermore,
we are developing a more compact TCS device that can bemounted on actual robots.

1070

[2]

[3]
[4]

[5]

[6]
[7]

[8]

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