2008 8th IEEE-RAS International Conference on Humanoid Robots
December 1 ~ 3, 2008 / Daejeon, Korea

Vibration Damping Control of Robot Arm Intended
for Service Application in Human Environment
Dzmitry Tsetserukou, Naoki Kawakami, Susumu Tachi
University of Tokyo, Japan, dima_teterukov@ipc.i.u-tokyo.ac.jp, {kawakami, tachi}@star.t.u-tokyo.ac.jp

Abstract— The paper focuses on vibration damping control of a
new anthropomorphic robot arm enabling the torquemeasurement in each joint and tactile area recognition to ensure
safety while performing tasks of physical interaction withenvironment or human beings. The vibrations in heavily loaded
joints have risen due to compliances introduced into each joint of
the robot arm by means of torque sensors. Such oscillations
deteriorate the performance of the manipulator and present
undesirable disturbance during interaction with human beings.
In order to overcome this disadvantage we have proposed robust
technique allowing considerable reduction of the oscillation
amplitude through acceleration signal feedback. The detailed
design procedure of the lead controller enabling improvement oftransient response is presented in the paper. The experimental
results of vibration damping control show the feasibility of
proposed approach to suppress the oscillations of compliant
robot joint.

I. INTRODUCTION
The service robotics aimed at assistance to human ineveryday life environment is nowadays gaining increased
interest of researchers and industry. Such robots areprogrammed to perform continuously changing tasks inunstructured human being environment. Both coexistence,
when the human and robot share the same workspace, andcooperation, when the human and robot work on the same task,
impose strict requirements on manipulator behavior andcontrol in order to ensure safe interaction with environment,
effectiveness of target task execution, and ergonomic
cooperation.
Several effective collision avoidance strategies weredeveloped. For instance, the high-speed vision system
attached to the robot arm allows real-time collision avoidance[1]. Is should be noted, that such tasks under humansupervision as transporting the object, leading the robot tip via
force-following, performing the assembling tasks, require theprocessing algorithm of contact state.
Generally, twotechniques of safe interaction with robot are well recognized,
namely, passive compliance control and active force control.
Whereas the mechanical leaf springs [2] and variable
stiffness actuators [3] integrated into each joint allow
achieving fast response to external disturbance, they also
cause vibrations destabilizing the dynamic behavior.
Additionally, the robot control in such cases is complicated bymany unknown parameters (e.g. actuator stiffness anddamping). Another approach to improvement of the intrinsic
safety of robot is to reduce the weight of components. The
main drawback of this method is occurrence of undamped
structural vibrations.

978-1-4244-2822-9/08/$25.00 ©2008 IEEE

Opticaltorquesensors

Position, velocitya) Operator cockpit

Tactilesensors

Accelerometerb) Master armc) Developed whole-sensitive robot arm

Fig. 1 Robot teleoperation system

Active force control implies that the robot structure is stiff
enough to provide high position accuracy; and contact ability
with environment is achieved by excessive sensory system
feedback [4]. However, due to the time lag induced bysampling process, such robots pose dangerousness at the
moment of impact.
To realize the safe physical contact of entire robot arm
structure with human and to guarantee the collision avoidance,
we developed a whole-sensitive robot arm (by using artificial
skin and distributed torque sensors in each joint [5]). Whencontact with environment occurs, manipulator automaticallygenerates compliant motion according to the measured
external force vector (active control). Thus, the whole
structure of the manipulator can safely interact with an
unstructured
environment.
The
newly
developed
anthropomorphic manipulator having 4-DOF arm and 8-DOF
hand (Fig. 1(c)) is capable of safe interaction withenvironment wherever contact occurs on the arm surface [6].
The torque sensors introduce compliances, which soften theimpact forces on the initial state of contact transient (passive
control). The teleoperated robot arm is controlled byoperator’s motion (Fig. 1(a)). During operation of manipulator
the movement information (position, orientation, and velocity)
is acquired through encoders of the motors installed into eachjoint of the master arm (Fig. 1(b)). The motion of the elbow iscontrolled by the tilt sensor.

On the other hand, joint flexibility increases the vibration
during acceleration and deceleration stages because of inertial
loading. Thus, the main goal of this paper is elaboration of thevibration damping control algorithm to achieve the smooth
continuous motion while interacting with environment.
The remainder of the paper is structured as follows. Section
2 introduces the mathematical model of the robot joint withflexibility, and method for oscillation parameter identification.
The lead compensator design algorithm is presented in Section
3. The experimental results are given in Section 4. In Section
5, we briefly conclude the paper and discuss the future work.
II. THE MODEL OF ROBOT WITH JOINT FLEXIBILITY
A. Dynamic Model of Flexible Robot Joint
Flexibility introduced into robot arm joint throughharmonic drive and torque sensors changes the dynamic
behavior of mechanical structure. Robot joint with compliancecan be represented as three-inertia system (Fig. 2) including:(1) the inertias of motor Jm, harmonic drive output shaft JD,
and load JL; (2) the spring elements of harmonic drive (withtorsional stiffness k1) and torque sensor (with torsional
stiffness k2).

Jm

J1, Pitch
6.1

6.1

3.33

3.33

Torsional stiffnessof the torquesensors k2,[kNm/rad]

3.23

2.73

0.95

0.95

Equivalenttorsional stiffnesske, [kNm/rad]

2.112

1.886

0.739

0.739

The dynamic equations of the n-DOF manipulator withjoint compliance in joint space coordinates can be obtained by
D’Alembert’s principle. The equations of static equilibrium
for mass A (Eq. (2)) and mass B (Eq. (3)) ignoring damping
are as follows:()
J θ = −i k ( i θ − θ ) + τ

J L (θ L )θL = k i −1θ m − θ L + τ dk1

τ = k (i θm − θL ) ,Due to neglible value of inertia of output shaft of harmonic
drive, three-inertia model can be reduced to the equivalent
two-DOF one with mass inertias i2Jm and JL (Fig. 3).

θL= 0i -1θm
i Jm
Tb

k

L

m

(2)
(3)
(4)
(5)

where θ L , θL , θL ∈ Rn are the vectors of the link angles, theFig. 2 3-DOF vibration system

JL

m

−1

k2

2

−1

τ d = τ EXT − G (θ L ) − C (θ L ,θL )θL

JL

Elbow
J4, Pitch

Torsional stiffnessof harmonic drive
k1, [kNm/rad]−1i

Arm joints
ShoulderJ2, Roll
J3, Yaw

Parameters

m m

JD

i τm

TABLE ITORSIONAL STIFNESSES OF ROBOT JOINTS

θL

τd

Ta

B
A
Fig. 3 Equivalent 2-DOF vibration system

The torsional stiffnesses k1, k2, and equivalent spring
constant ke derived from Eq. (1) are listed in Table 1.
1 1 1
kk= + , k = ke = 1 2 .
ke k1 k2
k1 + k2(1)

link angular velocities, and the link angle accelerations,
respectively; θ m ,θm ∈ Rn are the vectors of motor angles, andthe motor angle accelerations, respectively; k ∈ Rn×n is the
diagonal matrix of joint stiffness; i ∈ Rn is the vector of gear
reduction ratios; JL(θL) ∈ Rn×n is the symmetric, positive
definite link inertia matrix; Jm ∈ Rn×n is the diagonal, positive
definite rotor inertia matrix; τd ∈ Rn is the vector of
disturbance torques; τm ∈ Rn is the vector of torques appliedby motor; τEXT ∈ Rn is the vector of applied external joint
torques; G(θL) ∈ Rn is the vector of gravitational torques;C( θ L , θL ) ∈ Rn represents the vector of Coriolis andcentrifugal torques; τm ∈ Rn is the vector of torques applied by
motor; the torque in the shaft τ ∈ Rn is directly proportional
to twist angle θ = (i-1θm-θL).
The transfer function of two-mass robot joint can be
deduced from Eq. (2)-(5) and has the block diagram
presentation shown in Fig. 4. To provide the faithful tracing of
the motion of operator arm and to generate desired compliant
trajectory while interaction with environment, each joint has
PD control loop shown in Fig. 5 (where θ md ,θmd are thedesired angle and angular speed; Kp, Kv are proportional andderivative gains). The under-dumped response of motor
dynamics can be easily derived through PD gains adjustment.
However, there is no feedback signal from load position to
enable control of the unbalanced inertial loading dynamics.

1
i

τm

1
Jm s 2

θm

1
i

τd

τ

k

1
Jm s 2

Several techniques were proposed to attenuate vibrations byusing negative torque feedback [7], [8]. The idea behind thisapproach is that negative torque feedback reduces theeffective inertia of the motor rotor in (KT+1) times, where KT
is the torque gain. However, whereas this approach can
suppress the vibrations to some extent, it cannot control themain source of vibrations, namely, the inertial loading.

θL

Fig. 4 Block diagram of two-mass robot joint

θmd

τm

Kp

θmd

Kv

1 s 2 ωL2
Jm s 2 s 2 ωn2

θm

θm

ωL2
s 2 ω2L

1
i

θL

sfc
s + fcB. Oscillation Parameter Identification
The approach proposed by us is to employ position
feedback based on information from accelerometer and leadcontroller to improve the damping property of the compliant
robot joint. To illustrate the torsional vibrations caused byunbalanced inertial load (see the open-loop block of diagramin Fig. 5), let us consider dynamic model of robot arm given
in Fig. 7.

Fig. 5 Block diagram of two-mass robot joint with PDl

The natural frequency of two-mass system ωn and natural
frequency of one-mass robot joint ωL are derived from thefollowing equations:
⎛ 1
1 ⎞,
+
⎟
i
J
J
L ⎠
⎝ m

ωn = k ⎜ 2

ωL =

k .
JL

Target and real trajectory, (deg.)

Applied torque TEXT , (Nm)Joint angle θ1
Target impedance trajectory

-3
-6

1.5

3.0

4.5

6.0

Time t, (sec.)

7.5

9.0

1.0
0.5
0.0

Fig. 7 Graphical presentation of torsional vibrations

The desired trajectory generated by master arm movement
or, in case of interaction with environment by admittancecontroller, is described by angle θd. The flexibility of the joint
produces vibrations about Z-axis of corresponding joint. Using
the balance equations of forces acting on the joint we canwrite:(7)-1.0

Vibrations at deceleration stage of motion-1.5

Actual data plot
Low-Pass filtered data plot

1.5

3.0

4.5

6.0

Time t, (sec.)

acceleration of the link, respectively; b is a viscous damping
constant. We can neglect the influence of gravitational force
due to small vibration angle change.
By Laplace transform we get the following transfer
function:θ L ( s)
ωL2
k
,
=
=
θ d ( s ) J L s 2 + bs + k s 2 + 2ζωL s + ωL2

-0.5

b)

m

where θ L , θL , θL are actual angle, angular speed and angular

1.5

-2.0
0.0

b

J LθL + bθL + k (θ L − θ d ) = 0 ,

-9

a)

aτ

k

aτ

3

-12
0.0

θ
θd

(6)

The experimental results of human-robot interaction for the
first joint show that despite the smooth trajectory (Fig. 6(a))
generated by the admittance controller according to the sensed
joint torque, the vibratory behavior is recorded by torquesensor (Fig. 6(b)).
0

r

(8)

where ζ is the damping coefficient which can be derived fromthe following equation:
7.5

9.0

Fig. 6 Experimental results of human-robot interaction

(9)The oscillation periods of the angle of link, angularacceleration, and angular speed are identical. Therefore, fromanalysis of the angular acceleration signal we can obtain theunknown parameters of the joint, namely: natural frequency
and damping coefficient. The linear tangential acceleration aτ
can be measured by accelerometer. Then, the corresponding
angular acceleration is given by:

α = aτ r ,

20

2

b .
2 kJ L

Angular acceleration α, (rad/s )

ζ =

15

α = 15.142

10

α = 1.696

5
0
-5
-10

T = 0.106 sec

-15
-20
0.0

(10)

where r is the radius of curvature or distance to theaccelerometer (in our case it equals 0.241 m).
The three-axis accelerometer with sensitivity of 600 mV/G
was used. The special plastic case was manufactured in order
to fixedly attach the body of the sensor to the tip of forearm.
The model of accelerometer attachment along with coordinate
systems associated with robot elbow joint and sensor are
given in Fig. 8.

Actual data plot
Low-Pass filtered data plot

0.5

1.0

Time t, (sec)

The damping coefficient can be calculated from:

ζ =

δ=
X4

Za

r
HolderXa

Tip of the forearm

Cover

Fig. 8 Accelerometer attachment scheme

As seen from Fig. 8, the angular acceleration corresponds
to the X component of the accelerometer output. Duringexperiment, elbow joint was fixedly mounted, and the impulse
force was applied to the tip of the robot forearm. The
vibrations were visually observed. Fig. 9 shows the recorded
acceleration curve. To eliminate high frequency noise, thedigital low-pass filter was deployed with cut-off frequency of
100 Hz.
The procedure of oscillation parameter determination ispresented below.

4π + δ
2

,

(11)

2

1 ⎛ αn ⎞ ,ln ⎜⎟m ⎝ α n+m ⎠(12)where αn, αn+m are the measured amplitudes of angularacceleration; m is the number of peaks over which theamplitude change is measured [9].
To estimate the logarithmic decrement reliably, the numberof peaks was assigned to 14. The calculated value of
logarithmic decrement δ is 0.1564, and damping coefficient ζis equal to 0.0249 that indicates that system is lightly damped.
The damped natural angular frequency ωd is given by:ωd =Accelerometer
Coordinate system associatedwith accelerometerδ

where δ is the logarithmic decrement expressed by:

J4

Ya

2.0

Fig. 9 Angular acceleration plot

Y4
Z4

1.5

2π
2π
=
= 58.28 rad/sec.
0.106
T

(13)

Then, the natural angular frequency is calculated as:

ωL =

ωd
1− ζ

2

=

58.28
1 − 0.0249 2= 58.3 rad/sec.

(14)

After substitution of calculated parameters into Eq. (8), thetransfer function becomes:

θ L (s)3398.9.
= 2
θ d ( s ) s + 2.903s + 3398.9(15)

To reduce the lightly damped vibrations, the oscillationdamping controller is, therefore, needed.
III. DESIGN OF VIBRATION DAMPING CONTROLLER
We are proposing to use load angle position feedback andlead compensator to damp the undesirable vibrations. The
angular position can be calculated from accelerometer data bymeans of double integration. The high-pass filtering isnecessary to eliminate the drift of the signal (Fig. 10). The
cut-off angular frequency was assigned to 3 rad/sec.

θL

1
s2

s2
(s ωHP) 2

ωL2
s 2 2ζωLs ωL2

θL

Fig. 11 System with lead compensator

First, we determine the gain K to satisfy the steady-state
error ess (less then 1.0 % for a unit step function):

The transfer function is expressed as:y(s) θ L (s) 1
s2
1
.
= 
= 2
=
2
2
u ( s ) θ L ( s ) s ( s + ωHP )( s + ωHP )(16)

1 ⎤ ⎡θ L (t ) ⎤ ⎡ 0 ⎤⎡θL (t ) ⎤ ⎡ 0
⎢
⎥=⎢ 2
⎥⎢
⎥ + ⎢ ⎥ u (t )⎣ v(t ) ⎦ ⎣ −ωHP −2ωHP ⎦ ⎣ v(t ) ⎦ ⎣1 ⎦⎡θ (t ) ⎤y (t )= [1 0] ⎢ L ⎥ ,⎣ v (t ) ⎦

(17)

(19)
(20)

The matrices Ad and Bd are given by:

)

(21)
(22)

2ζ

(24)

.

−2ζ + 1 + 4ζ(23)

where ωz, ωp are the zero and pole; K is the constant gain.
The block diagram of the closed loop system with leadcompensator is shown in Fig. 11.

(25)

2

The derived phase margin φ is 55.7°. Value of theuncompensated system phase margin is as low as 0.29°.
Therefore, we need to add phase lead, which satisfies required
damping property of the system. With account ofcompensation of shift in the gain crossover frequency the
maximum required phase lead angle φmax is approximately 60°.
The coefficient α is computed from:α=

1 − sin(φmax )1 + sin(φmax )

.

(26)

The new gain crossover frequency ωc for total compensatedsystem is given by intersection of line 20 log10 a and bodemagnitude plot (for α of 0.0718, ωc=-11.44 dB). The final stepis to calculate corner frequencies ωz, ωp. Taking advantage ofthe fact that the maximum phase lead angle φmax occurs at thegeometric mean of two corner frequencies ωz, ωp, we can get:

ω z = ωc α ,To improve the system performance and to meet theperformance specification, we should design the compensator.
Such properties of lead compensator as speeding the response
up and increasing the stability of the system make it thepreferable one. The compensator contributes a phase lead atexisting gain crossover frequency. The transfer function oflead compensator [10] is written as:
K ( s + ωz ) K ( s + ωz ) ,=α ( s + ω p ) α ( s + ωz α )φ = tan −1(18)

the signal from accelerometer; and y(t) is the output signal. To
implement program code we have to use the discrete
presentation of the functions with sampling time of T. For
calculation step k we have:⎡ e −ωHPT + ωHPTe −ωHPT⎤
Te −ωHPT
Ad = ⎢− ωHPT−ωHPT−ω HPT ⎥2+ ωHPTee⎣ −ωHPTe⎦−ω HPT
−ωHPT
2
⎡ −e
⎤
− ωHPTe+ 1 ωHP
Bd = ⎢⎥.
− ωHPTTe⎣⎢⎦⎥1≤ 0.01 .
1 + Gc (0)G (0)2where v = θL is state variable; the input of the system u(t) is⎡θ L ( k +1) ⎤⎡θ L ( k ) ⎤⎢θ⎥ =Ad ⎢θ ⎥ + BdU ( k )⎣ L ( k +1) ⎦⎣ L(k ) ⎦.
y( k ) = θ L ( k )ess =The calculated value of K is 99. The specification of system
damping coefficient of 0.55 provides the value of required
phase margin φ defined by:State-space presentation is described by:

Gc ( s ) =θdK ( s ωz )α (s ωp)

θL

Fig. 10 Diagram of acceleration processing(uω p = ωcα.

(27)

The values of ωz and ωp equal 321 rad/sec and 4477. Using
the above mentioned procedure, the discrete presentation of
the lead compensator with sampling time of T is:x( k +1) =e

−ω pT

(

x( k ) + (1 − e

−ω pT

)

) ω p U c(k )y( k ) =θ d ( k ) =(ω z -ω p )x( k ) + U c ( k ) ,

(28)
(29)

where Uc(k) is the discrete input signal.
The obtained value of desired load angle θd is thenprocessed by PD control loop. The magnitude curve andphase-angle curve for the compensated system are shown in
Fig. 12. The phase and gain margins are 60 deg. and +∞,respectively. Therefore, compensated system meets both thesteady-state and the relative stability requirements.

Bode Diagram

resists the quick change in angular speed, and, therefore,
provides more smooth dynamic behavior. It is important toemphasize here, that theoretically calculated control gain andfeedback gain must be adjusted while experiment conducting.
Too high gains can cause increasing sensitivity to noise signal.
The signal drift presented in Fig. 13(b) is generated by smallchange in angular balance position.

Magnitude, (dB)

80
40
0
-40
-80

Phase, (deg)0-60

φ = 60 deg at ωc=1067 rad/sec

-120
-180
1

10

2

10

3

10

Frequency, (rad/sec)
Fig. 12 Bode diagram

4

5

10

10

Thus, the desirable transient behavior was achieved throughusage of lead compensator. To verify the theory and toevaluate the feasibility and performance of the proposed
damping controller, the experiment was conducted.

20

2Angular acceleration α, (rad/s )IV. EXPERIMENT AND DISCUSSION
During the experiment the impulse force was applied to thetip of the forearm, and the acceleration signal with desired
angle was recorded (Fig. 13).
15

Actual data plot
Low-Pass filtered data plot

105

V. CONCLUSION AND FUTURE RESEARCH
The flexibility of robot joint caused by compliant elements,
such as torque sensors, harmonic drives, and lightweightmechanical connectors, produce significant effect on the
dynamics of robot. The presented experimental results prove
oscillatory behavior of joints of whole sensitive robot arm.
To compensate flexibility effect, first, we elaborated the
dynamic model of the robot joint as two-mass system. The
system with lead compensator and angular position of the load
feedback is proposed to dump undesired vibrations. The
oscillation parameters required to design the lead controller
were determined by using experimental data and analyticalequations. The experimental results show high performance ofthe developed controller in terms of successful damping ofvibrations. In the future research we will address the tasks of
implementation of vibration damping in each robot joint andrealization of adaptive compensator to adjust parameters
according to the effective inertia.
ACKNOWLEDGMENT
The research is supported in part by a Japan Society for the
Promotion of Science (JSPS) Postdoctoral Fellowship for
Foreign Scholars.

0

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-5
[1]

-10
-15
-20

[2]

-25
-30
0.0
a)

0.5

1.0

1.5

2.0

Time t, (sec)
[4]

1.5

Desired angle θd , (deg.)

[3]

1.0

[5]

0.5

Signal drift

0.0

[6]

-0.5
-1.0

[7]

-1.5
b)

0.0

[8]

0.5

1.0

Time t, (sec)

1.5

2.0

Fig. 13 Experimental results

The experimental results show the successful damping ofoscillation magnitude. As follows from Fig. 13(b), the system

[9]
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