SICE Annual Conference 2008
August 20-22, 2008, The University Electro-Communications, JapanThe Control Algorithm Enabling Natural Physical Interaction with
Humanoid Robot Arm Having Joint Flexibility
Dzmitry Tsetserukou, Naoki Kawakami, and Susumu Tachi
Department of Information Physics and Computing, University of Tokyo, Tokyo, Japan(Tel : +81-3-5841-6917; E-mail: dima_teterukov@ipc.i.u-tokyo.ac.jp)
Abstract: The paper is concentrated on control of humanoid robot arm intended for operation in unstructured humanbeing-environment. A variable impedance control strategy was elaborated to increase the robot functionality and toachieve human-like dynamics of interaction. The detailed design procedure of the lead controller enabling improvementof transient response is presented. The experimental results show that proposed approach not only provides safeinteraction of robot arm with a person, but also improves the effectiveness of contact task performance.
Keywords: Variable impedance control, safe human-robot interaction, joint flexibility, teleoperation.

1. INTRODUCTION
In designing the human-friendly robots intended forphysical cooperation, the primary concern is safetyissues. Unexpected collisions should be managed insuch a way that contact forces do not pose a danger forhuman beings. Generally, two techniques of safeinteraction with robot are well recognized, namely,
passive compliance control and active force control.
Passive compliance of robot can be achieved byintegration of mechanical leaf springs [1] or variable
stiffness actuators [2] into each joint gaining fast
response to external disturbance. However, they also
can cause vibrations with high amplitude thatdestabilize the dynamic behavior. Additionally, the robot
control in such cases is complicated by many unknown
parameters (e.g. actuator stiffness and damping).
Pneumatic muscles with low weight and inherent
compliant behavior due to air compressibility are also
exploited in human-friendly robots [3]. The limitationsof such actuators are nonlinear flexible characteristics,
long reaction time, and small position accuracy.
Active force control implies that the robot structure isstiff enough to provide high position accuracy; andcontact ability with environment is achieved byexcessive sensory system feedback [4]. However, due tothe time lag induced by sampling process, such robots
pose dangerousness at the moment of impact.
In our robot design we embodied advantages of bothmethods. The developed anthropomorphic manipulator
having 4-DOF arm and 8-DOF hand (Fig. 1(c)) is
capable of safe interaction with environment. To realize
the safe physical contact of entire robot arm structurewith human, the developed optical torque sensors were
distributed into each joint of the robot arm. Whencontact with environment occurs, manipulator
automatically generates compliant motion according to
the measured external torque values (active control). On
the other hand, the torque sensors designed with low
stiffness mitigate the impact forces on the initial state ofcontact transient (passive control).
The teleoperation system aimed at performing thecooperative tasks with humans is presented in Fig. 1.

Opticaltorquesensorsa) Operator cockpit.

Position, velocity

Accelerometerb) Master arm.

c) Developed robot arm.

Fig. 1.Robot teleoperation system.
The robot arm is controlled by operator’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
each joint of the master arm (Fig. 1(b)). The motion of
the human elbow is controlled by the tilt sensor.
There are several conflicting requirements on the
choice of dynamics parameters of impedance model toprovide effectiveness and functionality of robot in tasksof physical interactions. For example, whileaccomplishing service tasks for human in theautonomous mode, it is required to provide highstiffness (to ensure small position error during object
handling) and high damping (for good velocity tracking).
In the case of collision, the low stiffness is necessary to
reduce the impact forces. We elaborated a new
methodology for joint impedance parameter regulation
based on human-like ability to softly interact withenvironment.
The paper also focuses on the vibration damping
control to attenuate oscillations in heavily loaded joints
during acceleration and deceleration stages. The
algorithm providing smooth continuous motion whileinteracting with environment is detailed. Experimental
results of variable impedance control and vibration
damping control are discussed.

- 2777 -PR0001/08/0000-2777 ¥400 © 2008 SICE

2. JOINT IMPEDANCE CONTROL
2.1 Impedance control
To achieve skillful human-like behavior, the robot has
to be able to change its dynamic characteristics
depending on time-varying interaction forces. The mostefficient method of controlling the interaction between amanipulator and an environment is impedance control.
This approach enables to regulate response properties ofrobot to external forces through modifying themechanical impedance parameters (Fig. 2).

FEXT
Jdi

Dd(i+1)Jd(i+1)W EXT (i+1)

Ddi
Kd(i+1)
WEXTi

Kdi

Fig. 2 Presentation of the local impedance control.

2.2 Variable joint impedance control
The limitation of impedance control with constantcoefficients is that it cannot provide adaptation of robot
dynamic behavior to different tasks. To overcome this,two approaches were developed: adaptive [5] andfunctional [6] adjustments of impedance parameters. We
have proposed a new method based on human-likeability to softly interact with environment.
It was found that humans adjust their joint stiffness toaccommodate changes in surface stiffness [7].
Furthermore, the research on impedance characteristics
of human arm shows that, while pushing the object
naturally, human arm stiffness and damping behavior
can be approximated by exponential curves [8].
To impart the human-like damping and stiffness
behavior to robot arm interacting with environment, we
introduce the exponential functional dependency
between sensed force and stiffness. For service task
accomplishment (when high stiffness and damping of
joints is required), the thresholds of external disturbancetorque WEXTth was assigned for each joint of robot arm.
The desired stiffness Kd1 for the case of service task is
calculated based on the maximum deflection of joint
angle 'Tmax caused by external torque WEXT:

The desired impedance properties of i-th joint ofmanipulator can be expressed as:
J di 'și  Ddi 'ș i  K di 'și

Ĳ EXTi ; 'șiș ci  ș di ,(1)

where Jdi, Ddi, Kdi are the desired inertia, damping, andstiffness of i-th joint, respectively; WEXTi is torque applied
to i-th joint, 'Ti is the difference between the current
compliant angle Tci and desired one Tdi.
The state-space presentation of the Eq. (1) is:ª 'ș i º ª 0« »=«¬ ȣ i ¼ ¬  K d J d1º ª și º ª 0 º
Ĳ (t ) , Dd J d ¼» ¬« ȣi ¼» ¬«1 J d ¼» EXTi

(2)

where Xi is the state variable (ȣi 'ș i ).
The discrete presentation of Eq. (2) with samplingtime of T is needed for program code implementation:ª 'ș k +1 ºª 'ș k º« 'ș » =Ad « 'ș »  Bd TEXT ( k ) .
¬ k +1 ¼¬ k¼(3)

For the fastest non-oscillatory response on theexternal force (when the eigenvalues of matrix A O1 andO2 are real and equal), we have:Ad
BdTª 1  Ȝ1Tº,eȜ1T «»¬  K d T / J d 1  Ȝ1T  Dd T / J d ¼
Ȝ1T
1 ª 1  e (1  Ȝ1T ) º ,Ad  I A1 B«»
K d ¬« Dd /(2 J d ) 2 TeȜ1T ¼»

where I is the identity matrix.

(4)

K d1

Ĳ EXT .
'ș max(6)

It was defined that external torque of 1 Nm results in'Tmax of 0.1 rad yielding Kd1 of 10 Nm/rad. The desired
damping Dd1 is expressed as:
Dd 1

(7)

2ȗ K d 1 J d 1 .

Damping coefficient ] of 1.05 and desired inertia Jd1of 0.1 kgm2 was defined to achieve fast non-oscillatory
response giving Dd1 of 2.1 Nms/rad. These parameters
are valid till the interaction force does not cause theoverload of robot arm. When sensed torque value
exceeds the threshold, robot recognizes this condition as
collision or human following motion mode, and adjusts
its dynamics parameters to provide smooth naturalinteraction. To realize such continuous change ofdynamics, we are using exponential relation between
external disturbance torque and desired stiffness:

Kd 2

K d 1eȝ Ĳ EXT  Ĳ EXTth,(8)

where Kd2 is the variable desired stiffness aimed at softinteraction; P is the joint stiffness reduction coefficient.
The desired damping is adjusted to prevent force
responses from being too sluggish while changing
stiffness values, and to ensure contact stability:

(5)

Dd 2

2ȗ K d 2 J d 12ȗe

0.5ȝ Ĳ EXT  Ĳ EXTthK d1J d1 .

(9)Then, Eq. (1) for variable joint impedance controller
is written as:- 2778 -Ĳ EXTi

J di 'și  2ȗe

 K di1e

ȝ Ĳ EXTi  Ĳ EXTthi

0.5ȝ Ĳ EXTi  Ĳ EXTthi'și ; 'șiK di1 J di1 'ș i (10)ș ci  ș di .

To ensure the effectiveness of service task
accomplishment, we decided to implement admittancecontrol. In this algorithm, compliant trajectory
generated by the impedance controller is tracked by the
PD control loop. During the experiment, the interaction
with robot arm was performed to exceed join torquethreshold level of 0.6 Nm. The experimental results forthe elbow joint – applied torque, stiffness and damping,
impedance trajectory with constant and variable
coefficients – are shown in Figs. 3(a)~3(d), respectively.
External torque WEXT , Nm

0.8

0.4

WEXTth6(P=-1.155)

Stiffness, Kd (Nm/rad)Dampind, Dd (Nm s/rad)

4

0.2

2

Impedance trajectory 'Tk+1 , deg

24Time t, sec

6

8

0

-2

2

4

6

Time t, sec

8

䎃

0

2

4

Time t, sec

6

8


J L (ș L )șL

Varible coefficients
Constant coefficients

d)

0

2

4

6

Time t, sec

810

Fig. 3 Experimental results.

Applied torque WEXT , Nm

1.5
1.0
0.5
0.0

-0.5

Vibrations at deceleration stage of motion
Actual data plot
Low-Pass filtered data plot

-1.5
-2.0
0.0

1.5

3.0

4.5

6.0

Time t, sec

k i 1ș m  ș L  Ĳ d

(11)

J m
șm

i 1k i 1ș m  ș L  Ĳ m

(12)

Ĳd

Ĳ EXT  G (ș L )  C (ș L ,ș L )ș L

(13)

Ĳ

k i 1ș m  ș L ,(14)

where ș L ,ș L ,șL  Rn are the vectors of the link angles,

The plots represented in Figs. 3(c), 3(d) show that the
proposed variable impedance control provides smoothertrajectory while avoiding impact than the impedancecontrol with constant coefficients.

-1.0

Wd

The dynamic equations of the n-DOF manipulator
with joint compliance in joint space coordinates can beobtained by D’Alembert’s principle. The equations ofstatic equilibrium for mass A (Eq. (11)) and mass B (Eq.
(12)) ignoring damping are as follows:

0

-8

Ta

k

A
Fig. 5 Two-inertia vibration model of robot joint.

-6

Variable K, D
Constant K, D

-8

TL

B

-4

-6

Tb

i Wm

-2

-4

c)

b) 0

Impedance trajectory 'Tk+1, deg

0.0

a) 0

-10

i Jm

8
Threshold torque

JL

2

10

1.0

0.6

harmonic drive and torque sensors changes the dynamic
behavior of mechanical structure. Robot joint withcompliance can be represented as two-inertia system(Fig. 5).
T=0Li -1 Tm

7.5

9.0

Fig. 4 Vibrations in heavily loaded joint.
However, experimental results of human-robot
interaction for the inertially loaded first joint show that
despite the smooth trajectory generated by theadmittance controller, the vibratory behavior is recorded
by torque sensor (Fig. 4).

3. THE MODEL OF ROBOT WITH JOINTFLEXIBILITY
3.1 Dynamic model of flexible robot joint
Flexibility introduced into robot arm joint throughthe link angular velocities, and the link angle  Rn are the vectors
accelerations, respectively; ș m ,șmof motor angles, and the motor angle accelerations,
respectively; k  Rnun is the diagonal matrix of joint
stiffness; i  Rn is the vector of gear reduction ratios;
JL(TL)  Rnun is the symmetric, positive definite linkinertia matrix; Jm  Rnun is the diagonal, positive
definite rotor inertia matrix; Wd  Rn is the vector of
disturbance torques; Wm  Rn is the vector of torques
applied by motor; WEXT  Rn is the vector of appliedexternal joint torques; G(TL)  Rn is the vector ofgravitational torques; C (ș L ,ș L )  Rn represents thevector of Coriolis and centrifugal torques; Wm  Rn is thevector of torques applied by motor; the torque in theshaft W  Rn is directly proportional to twist angle T =(i-1Tm - TL).
The natural frequency of two-mass system Zn andnatural frequency of one-mass robot joint ZL are derived
from the following equations:
Ȧn

§ 1
1 ·
k¨ 2  ¸,© i Jm JL ¹

ȦL

k .
JL

(15)

The damped response of motor dynamics can beachieved by D-gain augmentation of position controller.
The technique of vibration attenuation by using negative- 2779 -torque feedback was reported [9,4]. The idea behind thisapproach is that negative torque feedback reduces theeffective inertia of the motor rotor. However, whereas it
can suppress the vibrations to some extent, it cannot
affect the main source of vibrations, namely, the inertial
loading. To design the vibration damping controller theoscillation parameter determination is needed.

given in Fig. 7.

3.2 Oscillation parameter identification
The approach proposed by us is to employ position
feedback based on information from accelerometer andlead controller to improve the damping property of thecompliant robot joint. To illustrate the torsional
vibrations caused by unbalanced inertial load, let us
consider dynamic model of robot arm given in Fig. 6.

Za

Y4
J4

Z4

Ya

X4

r

Forearm frames
Xa

Ya

Tip of the forearm
Holder
Accelerometer

Za

Cover

Xa
Coordinate system associatedwith accelerometer

Fig. 7 Accelerometer layout.

k

As seen from Fig. 7, the linear tangential accelerationaW corresponds to the X component of accelerometer
output. Angular acceleration of robot link equals aW/r.
While dynamic testing, elbow joint was fixedly
mounted and the system was exited by impulse force.
The data acquisition system recorded the subsequentvibrations with a sampling rate of 100 kHz (Fig. 8). To
eliminate high frequency noise, the digital low-passfilter was deployed with cut-off frequency of 100 Hz.

aW
b

m

2

aW

Fig. 6 Graphical presentation of torsional vibrations.
The desired trajectory generated by master arm
movement or, in case of interaction with environment
by impedance controller, is described by angle Td. The
flexibility of the joint produces vibrations about Z-axisof corresponding joint. We can neglect the influence ofgravitational force due to small vibration angle change.
Using the balance equations of forces acting on the joint
we can write:
J Lș L  bș L  k ș L  ș d

0,k
J L s 2  bs  k

Ȧ 2L,
s 2  2ȗȦ L s  Ȧ 2L

D = 15.142

15Actual data plot
Low-Pass filtered data plot

10

D = 1.696

5
0
-5
-10

T = 0.106 sec

-15
-20
0.5

1.0

Time t, sec

1.5

2.0

Fig. 8 Impulse response of the robot link.
The calculated damping coefficient ] of 0.0249 andnatural frequency of robot link ZL of 58.3 rad/sec
indicate that system is lightly damped andoscillation-damping controller is, therefore, needed.

4. DESIGN OF VIBRATION DAMPINGCONTROLLER(17)

where ] is the damping coefficient.
The oscillation periods of the angle of link, angularacceleration, and angular speed are identical. Therefore,
from analysis of the angular acceleration signal we canobtain the values of natural frequency and damping
coefficient. The three-axis accelerometer withsensitivity of 600 mV/G was used. The accelerometerlayout along with associated coordinate systems are

20

0.0

(16)

where ș L ,ș L ,șL are actual angle, angular speed andangular acceleration of the link, respectively; b is aviscous damping constant.
By Laplace transform we get the following transfer
function:ș L ( s)ș d (s)Angular acceleration D, rad/s

TL
Td

r

We are proposing to use load angle position feedback
and lead compensator to damp the undesirable
vibrations. The angular position can be calculated fromaccelerometer data by means of double integration. The
high-pass filtering with cut-off angular frequency ZHP of
3 rad/sec is necessary to eliminate the drift of the signal.
The transfer function of double integrator withhigh-pass filter is expressed as:- 2780 -1s2
2
s s  Ȧ HP 21s  Ȧ HP

2

.

(18)

To improve the system performance and to meet theperformance specification, we should design thecompensator. Such properties of lead compensator as
speeding the response up and increasing the stability ofthe system make it the preferable one. The transfer
function of lead compensator is written as:y(s)uc ( s )Gc ( s )K s  ȦzĮ s  ȦpK s  Ȧz ,
Į s  Ȧz Į

(19)

where Zz, Zp are the zero and pole; K is the constantgain [10].
The block diagram of the closed loop system withlead compensator is shown in Fig. 9.

uc

K ( s Zz )D (s Zp)

Td

ZL22s ]Zs
ZL2
L

TL

(20)

The calculated value of K is 99. The specification ofsystem damping coefficient of 0.55 provides the valueof required phase margin \ defined by:
ȥ

tan

2ȗ

1

2ȗ 2  1  4ȗ 2

.

(23)

Bode Diagram
80
40
0
-40
-80

\ = 60deg atZc=1067 rad/sec

-120

1

2

10

3

10

10

Frequency, (rad/sec)

4

10

5

10

Fig. 10 Bode diagram.
Thus, the desirable transient behavior was achieved
through usage of lead compensator.
From Eq. (19) the state-space representation of thelead controller is obtained:x (t )

Ȧ p x(t )  uc (t )(24)y (t )

K(Ȧ z  Ȧ p ) x(t )  uc (t ) .
Į(25)

where x(t) is the state vector, y(t) is the output, uc(t) is
the input.
Solution of the differential equation for x(t) is:x(t ) e Ȧ pttx(0)  ³ e

 Ȧ p ( t  Ĳ)

uc (Ĳ)dĲ .

(26)

0

For calculation step k and sampling time T we have:xk 1 =e

(22)

The new gain crossover frequency Zc for totalcompensated system is given by intersection ofline 20 log10 Į and bode magnitude plot (for D of
0.0718, Zc=-11.44 dB). The final step is to calculate
corner frequencies Zz, Zp. Taking advantage of the fact

-60

-180

(21)

The phase margin \ derived from Eq. (21) is 55.7q.
Value of the uncompensated system phase margin is as
low as 0.29q. Therefore, we need to add phase lead,which satisfies required damping property of the system.
With account of compensation of shift in the gaincrossover frequency the maximum required phase leadangle \max is approximately 60q. The coefficient D iscomputed from:1  sin(ȥ max ) .
Į1  sin(ȥ max )Į.

ȦcThe values of Zz and Zp equal 321 rad/sec and 4477
rad/sec.
The magnitude curve and phase-angle curve for thecompensated system are shown in Fig. 10. The phaseand gain margins are 60 deg and +f, respectively.
Therefore, compensated system meets both thesteady-state and the relative stability requirements.

First, we determine the gain K to satisfy thesteady-state error ess (less then 1.0% for a unit step
function):1
d 0.01 .
1  Gc (0)G (0)Ȧp0Fig. 9 System with lead compensator.

essȦc Į ,

Ȧz

Phase, (deg)uthat the maximum phase lead angle \max occurs at thegeometric mean of two corner frequencies Zz, Zp, we
can get:Magnitude, (dB)

ș L ( s)
ș L ( s)

y(s)
u (s)

 Ȧ pTyk =ș d ( k ) =) Ȧ p U c(k )(27)K(Ȧ z  Ȧ p )xk  U c ( k ) ,Į(28)

xk  (1  e

 Ȧ pT

where Uc(k) is the discrete input signal.
The obtained value of desired load angle Td(k) is thenprocessed by PD control loop (Fig. 11). To evaluate theperformance of the proposed damping controller, thedynamic testing was conducted.

- 2781 -Constant impedance
trajectory generator' Tk+1Td
Targettrajectory
generator

WEXT < W EXTthWEXT > WEXTth WEXT

Variable impedance
trajectory generator

Comparator

Tc

Td

Tc
Td

- +

Kv

+-

Wm

Kp

Wg

sfc
s + fc1 s 2 ZL2
Jm s 2 s 2 Zn2Actual data plot
Low-Pass filtered data plot

5
0
-5
-10
-15
-20
-25
-30
0.0

0.5

1.0

Tm

ZL2
s 2 ]Zs
ZL2
L

1
i

TL

1.5

Time t, sec
Fig. 12 Impulse response of system with controller.

2.0

It is important to emphasize here, that theoreticallycalculated control gain and feedback gain must be
adjusted while experiment conducting. Too high gains
can cause increasing sensitivity to noise signal.

5. CONCLUSIONS
A variable impedance control strategy was elaborated
to increase the robot functionality and to achieve
human-like dynamics of interaction. The effectivenessof the position-based variable impedance control for
providingsafe
human-robot
interaction
wasexperimentally illustrated on a whole-sensitive robot
arm. Tactile sensation of soft friendly interaction withrobot arm was achieved.
The flexibility of robot joint caused by compliant
elements, such as torque sensors and harmonic drives,
produces significant effect on dynamics of robot. To
compensate flexibility effect, first, we elaborated the
dynamic model of the robot joint as two-inertia system.
The oscillation parameters were determined by using
experimental data and analytical equations. The systemwith lead compensator and angular position of the load
feedback is proposed to damp undesired vibrations. The
experimental results show high performance of thedeveloped controller in terms of successful damping ofvibrations.

K (s Zz )E (s Zp)

Acknowledgments. The research is supported in partby a Japan Society for the Promotion of Science (JSPS)
Postdoctoral Fellowship for Foreign Scholars.

REFERENCES

20
10

Optical torquesensorA/D

Td

T

Experimental results show the successful damping ofoscillation magnitude (Fig. 12). The system resists thequick change in angular speed, and, therefore, provides
more smooth dynamic behavior.
15

T

Tc

Fig. 11 Block diagram of designed controller.

2-

Gravitycompensation'Tk+1 ' Tk+1

T

Angular acceleration D, rad/s

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- 2782 -