Proceedings of the 2004 IEEE
International Conference on Robotics & Automation
New Orleans, LA • April 2004

Impedance matching for a serial link manipulator
Ryo Kurazume

Tsutomu Hasegawa

Kyushu University
6-10-1, Hakozaki, Higashi-ku, Fukuoka, Japan
Email: kurazume@is.kyushu-u.ac.jp

Kyushu University
6-10-1, Hakozaki, Higashi-ku, Fukuoka, Japan
Email: hasegawa@irvs.is.kyushu-u.ac.jp

Abstract— We propose a new index for dynamic performance
analysis of serial link manipulators named Impedance Matching
Ellipsoid, or IME. Several indexes have been proposed forindicating static and dynamic performance of robot manipulators. For example, Dynamic Manipulability Ellipsoid (DME)characterizes distributions of hand acceleration produced bynormalized joint torque. Manipulating-Force Ellipsoid (MFE)
denotes static torque-force transmission ef£ciency from actuators
at joints to a hand. On the other hand, the proposed IME
characterizes dynamic torque-force transmission ef£ciency from
actuators at joints to a load held at the hand of the manipulator.
The IME includes a wide range of concepts proposed so far
as measures of manipulator’s performance. The DME and the
MFE are both derived from the IME as limiting forms about
the load mass. In this paper, we demonstrate the IME with somenumerical examples including the selection of an optimal leg
posture for jump robots, optimum active stiffness control, andan extension for manipulators mounted on satellites in outerspace.

I. I NTRODUCTION
A variety of indexes which indicate the performance ofrobot manipulators were proposed from the middle of 1980’s
to the beginning of 1990’s. For indexing the static performanceof robot manipulators, the Manipulability Ellipsoid (ME) [1]
was introduced indicating the relationship between angularvelocity at each joint and linear and angular velocity at a
hand of the manipulator. Several different indexes in this category were also introduced: The Manipulating-Force Ellipsoid(MFE) [2] indicates static torque-force transmission from thejoints to the hand; the compatibility index [3] for determining
an optimal posture for particular tasks; the condition numberof the Jacobian matrix. In addition, for indexing dynamic
performance of robot manipulators, the Dynamic Manipulability Ellipsoid (DME) [4],[2],[5] was proposed describing
distributions of possible hand acceleration. Another measure is
the acceleration radius [6], which is obtained as the minimum
value of upper boundaries of feasible hand acceleration intask space. Furthermore, the generalized inertial ellipsoid [7]which enables us to handle the distributed mass system of themanipulator as the single point mass system at the hand was
proposed.
These various indexes denote static and dynamic performance of manipulators themselves, speci£cally, in the casethat they are manipulated under a no-load condition. However,in some cases, we’d like to know the performance when they
carry loads in the hands. Especially, in the case that loads arelimited to a small variations such as pick-and-place tasks in

0-7803-8232-3/04/$17.00 ©2004 IEEE

assembly processes, the index which takes the effect of loads
into account is convenient.
For an actuator and gear system, the idea of “impedance
matching” has been widely used. This concept is usually
utilized for the choice of the optimum gear ratio in terms of thetransmission performance from torque produced at the actuator
to torque applied to the load. The £tness of inertial properties
between the actuator system including inertia of a rotor andthe load determines the performance of torque transmission.
On the other hand, the Jacobian matrix indicates the transformation between displacements of joint angles and positionof the hand of robot manipulators. Since this matrix can be
regarded to correspond with the gear ratio of the actuatorsystem mentioned above, it is possible to extend the conceptof “impedance matching” to serial link manipulators. This
new concept, namely, the impedance matching for serial link
manipulators, can be considered as the transmission ef£ciencyof torque produced at each joint to force applied to the loadat the hand of the manipulator. Its performance is determined
depending on the posture of the manipulator.
In this paper, we propose a new concept named ”theimpedance matching for serial link manipulators”[8] and a
new index for indicating manipulator’s dynamic performance
named Impedance Matching Ellipsoid, or IME. The proposed
IME characterizes the dynamic torque-force transmission ef£ciency from actuators at joints to the load held at the hand of
the manipulator. The IME includes a wide range of concepts
proposed so far as measures of manipulator’s performance.
The DME and the MFE, which have been considered asindexes which are based on distinct concepts [2], are bothderived from the IME as limiting forms about the load
mass. This paper demonstrates the use of the IME with two
numerical examples; the selection of an optimal leg posturefor jump robots and an extension to manipulators mounted onsatellites in outer space.
II. I MPEDANCE MATCHING FOR SERIAL LINKMANIPULATORS

A. Impedance matching for actuator systems
In this section, we introduce the basic idea of the impedance
matching for actuator systems. Let the moment of inertia of
the actuator including a rotor and a shaft be Im , the momentof inertia of the load be Il Cthe gear ratio be ξ, and the angularvelocity of the load be ωl as shown in Fig.1. The equivalent4802

torque produced by the actuator around the shaft, τm , can thenbe written as


Il
(1)
τm = Im + 2 ξ ω̇lξ

where ẍ ∈ RM is acceleration of the hand, Mp ∈ RM ×M isload mass, and g ∈ RM is acceleration of gravity. Moreover,acceleration of the hand can be obtained as
˙ q̇ẍ = J(q)q̈ + J(q)and the torque applied to the load, τl , is given asτl = Il ω̇l(2)

Therefore, the transmission ef£ciency from torque produced
at the actuator to torque applied to the load can be de£ned as
Im + ξI2l ξ ω̇l
τm
=
(3)
η=
τl
Il ω̇lThus, the optimum gear ratio that maximizes the transmission
ef£ciency obtained above is determined as

Il
ξ=
(4)
Im
By choosing the optimum gear ratio obtained in Eq.(4), a large
acceleration of the load is produced with a small output of
torque at the actuator. This idea is called as the impedance
matching of actuator systems.

Im
τm
ωl
Fig. 1.

Since load mass Mp is a regular matrix and by substituting
Eqs.(6) and (7) into Eq.(5), we get the motion equation interms of external force and moment as follows:τ˙ q̇)= M (q)J(q)† Mp−1 (Fe − Mp g − Mp J(q)+C(q, q̇) + G(q) + J(q)T Fe= Q(q)(Fe − Fbias )Impedance matching of an actuatorτ = M (q)q̈ + C(q, q̇) + G(q) + J(q)T Fe
N ×N

(5)

where τ ∈ R
is joint torque, M (q) ∈ R
is a tensor
of inertia of the manipulator in joint space, C(q, q̇) ∈ RN is
coriolis and centrifugal forces, G(q) ∈ RN is gravity force,J(q) ∈ RM ×N is the Jacobian matrix, and Fe ∈ RM is
external force and moment applied to the hand. Here, M shows
the degree of freedom of the manipulator hand.
On the other hand, the motion equation of the load held at
the hand is given asgFe = Mp ẍ + Mp
0
= Mp ẍ + Mp g
(6)

(8)

where
Fbias

=

(J(q)T + M (q)J(q)† Mp−1 )†˙ q̇) − C(q, q̇) − G(q)][M (q)J(q)† (g + J(q)(9)T

Q(q) = J(q)†+ M (q)J(q) Mp−1

(10)

and J(q)† ∈ RM ×N is a pseudo inverse of the Jacobian matrix
J(q). J(q)† can be obtained by the following equation in thecase that the Jacobian matrix is not a regular matrix.
J(q)† = W −1 J T (JW −1 J T )−1Il τl

B. Impedance matching for serial link manipulators
In this section, we extend the idea of the impedance matching of actuator systems to serial link manipulators. The derived
index indicates the ef£ciency of torque-force transmission
from actuators at each joint to the load held at the hand of
the manipulator. For illustrating the idea of the impedance
matching for serial link manipulators more clearly, ImpedanceMatching Ellipsoid, or IME, is also proposed.
The motion equation of serial link manipulators which
consist of N joints and N links is written as the following
equation.
N ×1

(7)

(11)

where W is a weight matrix.
Eq.(8) denotes the relationship between torque produced at
actuators and force and moment applied to the load from thehand. The coef£cient matrix, Q(q) ∈ R N ×M , indicates thetorque-force transmission ef£ciency, and F bias is a bias term
related to current velocity and gravitation. Thus, by applying
the singular value decomposition for this matrix Q(q), we get
Q(q) = U ΣV T

(12)

where Σ = diag(σ1 , σ2 , · · · , σM ) ∈ RM ×N , and U ∈RM ×M , V ∈ RN ×N are orthogonal matrixes. Therefore,transmission ef£ciency from torque produced at each joint toforce and moment applied to the load from the hand can be
described by the following equation.
w = σ1 · σ2 · · · σM

(13)

In this paper, we de£ne the value of w in Eq.(13) as an index
which indicates the degree of impedance matching for serial
link manipulators.
C. Impedance matching ellipsoid
In this section, we assume that torque limits at each actuatorare symmetrical, namely,−τilimit ≤ τi ≤ τilimit(14)

Let’s consider a conversion matrix, L
L = diag(τ1limit , τ2limit , · · · , τnlimit )(15)

Then, normalized joint torque is obtained with L as4803

τ̃ = L−1 τ(16)where τ̃ is normalized torque. Therefore, when the normalized
torque with the magnitude of 1 is produced, force and moment
applied to the load are derived fromτ̃ T τ̃ ≤ 1

Load
Hand

T

T

−2

(Fe − Fbias ) Q L

Q(Fe − Fbias ) ≤ 1

Cp

Mass-Spring-Damper model

(18)

Fbias

This equation indicates the ellipsoid of transmitted force andmoment from joints to the load, and thus, we de£ne thisellipsoid as Impedance Matching Ellipsoid, or IME.

=

The IME de£ned Eq.(18) coincides with Dynamic Manipulability Ellipsoid (DME) in the case that Mp → 0, andManipulating-Force Ellipsoid (MFE) in the case that Mp →∞. We will show this relation in this section.
At £rst, we assume that M p → 0. Then, since Fe = 0 from
Eq.(6), Eq.(8) can be rewritten as
˙ q̇) + C(q, q̇) + G(q)= M (q)J(q)† (ẍ − J(q)†= M (q)J(q) (ẍ − ẍbias )(19)(J(q)T + M (q)J(q)† Mp−1 )†[M (q)J(q)† (g + Mp−1 Cp ẋ + Mp−1 Kp ∆x˙ q̇) − C(q, q̇) − G(q)]
+J(q)
(24)

D. Features of the IME

τ

Mp

(17)
Fig. 2.

as

Kp

Q(q) = J(q)T + M (q)J(q)† Mp−1

(25)

Eq.(24) shows that spring and damping characteristics emerge
.
in the bias term Fbias
III. N UMERICAL EXAMPLES
A. Design of an optimum leg structure for jump robots
In this section, we introduce a numerical example of the
proposed impedance matching for determining an optimum
leg posture for a 2-link jump robot.

Thus, Eq.(18) becomes(ẍ − ẍbias )T (M (q)J(q)† )T L−2 (M (q)J(q)† )(ẍ − ẍbias ) ≤ 1(20)This equation is the de£nition of the Dynamic Manipulability
Ellipsoid.
Next, we assume that Mp → ∞. If we consider a stationary
state (ẋ = ẍ = 0) and ignore the effect of the gravity, Eq.(18)
can be revised as
FeT J(q)L−2 J(q)T Fe ≤ 1

E. Extension to Mass-Spring-Damper model
In some cases, the motion equation of the load held at thehand should be indicated by the following equation instead ofEq.(6) as shown in Fig.2 [9].
(22)

In this model, spring and damping characteristics at the contact point x are also considered. The proposed IME can take
into account of these contact conditions while the DME cannot
handle it. By taking these additional effects into consideration,
Eqs. (8), (9), and (10) are rewriten as the following equations.
)τ = Q(q)(Fe − Fbias

Load mass
φ2

1Kg
1Kg

z

φ1

x

2-dof manipulator(21)

This equation indicates the Manipulating-Force Ellipsoid.
Consequently, the IME is considered to have an intermediate
nature between the DME and the MFE, and the DME and the
MFE are both derived as limiting forms about the load mass.
Though the idea for integrating the DME and the MFE fora particular task was proposed by Koeppe [2], their idea wasbased on the selection of both indexes obtained individuallyin task space. On the other hand, the proposed IME is a novel
concept which includes both concepts of the DME and theMFE under the condition of a particular load mass.

Fe = Mp ẍ + Cp ẋ + Kp ∆x + Mp g

1Kg

(23)

Fig. 3.

Jumping robot

Let the length and the mass of each leg be 1[m] and 1[Kg],and the joint angles be φ1 and φ2 , respectively. Fig.4 depicts
the IME, the DME, and the MFE in the case that the bodymass is 1[Kg], φ1 = 45◦ , φ2 = 90◦ , and maximum torquelimit τ limit is 10[Nm]. The unit of the IME and the MFE is[N ] and the unit of the DME is [m2 /s].
Fig.4 shows that the IME is smaller than MFE since it
includes the torque consumption in order to drive the massof the manipulator itself. Moreover, the axis which indicatesthe highest torque-force transmission ef£ciency is given at theintermediate position between the one of the DME and theMFE. Thus, it can be said that the IME has an intermediate
feature of these ellipsoids in terms of its shape.
Next, we compare the feasible output force toward the z
direction for several initial postures. In this simulation, we
assume that the sole is always placed under the center of thebody, namely, φ2 = π − 2φ1 . Figs.5 and 6 depict the valuesof IME and the DME in the z direction for several initial
postures, respectively. Fig.5 shows that the proposed IME hasa peak in transmission ef£ciency at φ 1 = 0.95[rad]. On theother hand, the DME is monotonously decreasing as shown in
Fig.6 since it doesn’t take the body mass into account. Figs.7

4804

MFE (Manipulating-ForceEllipsoid)DME (Dynamic
Manipulability Ellipse)
IME z

IME (ImpedanceMatching Ellipse)

90

120

90

45

30

z

45

x

Fig. 4.

Fig. 7.

Impedance matching ellipsoid, IME

and 8 depict the IME and the DME for 4 different leg postures.
As straightening the legs, the distance of the DME in the z
direction decreases monotonously. But the IME increases its
distance at £rst, then begins to decrease.
The peak of the IME indicates the optimum posture in termsof torque-force transmission ef£ciency. Thus, this posturemakes it possible to apply large force to the body even if
the same joint torque is produced. Since the jumping heightis determined by the time integral of the force applied tothe body, the proposed IME indicates the performance ofthe link structure of the jump robot more correctly than theconventional indexes such as the DME or the MFE.
10

IME z [N]

8

6

45

30

Fig. 8.

60

75

IME for various leg postures

90

120

30

60

30

60

60

75

DME for various leg postureslinks are same as the model shown in Fig.3. In simulations,the torque of 300 [Nm] is applied to the joint 2 from 0[sec] to 0.1 [sec], then the joint 2 is £xed after 0.1 [sec]. An
example of simulation results in case of φ1 = 50◦ is shownin Fig.9. Maximum jumping height, which is the distancefrom the height when the sole leaves the ground, to the
maximum height, is depicted in Fig.10 for various leg postures.
Maximum jumping height has a peak in case of φ1 = 50◦
and it decreases rapidly if φ1 increases. These results arevery much in agreement with the expected results using the
proposed IME as shown in Fig.5.

4

2

0

0

10

20

30

40

50

60

70

80

90

φ1[deg.]Fig. 5.

Vertical force derived from IME(1) 0 sec

(2) 0.1 sec(3) 0.26 sec(4) 1.0 sec

10

DME z [m 2/s]

8

6

4

2

0

Fig. 9.
0

Fig. 6.

10

20

30

40
50
φ1[deg.]

60

70

80

An example of jumping simulation

90

B. Determinig optimum servo stiffness of active stiffness control for a serial link manuipulator

Vertical acceleration derived from DME

We conducted jumping simulations of a 2-link jump robot
using a dynamic simulator, OpenHRP. Length and mass of

The IME, DME, and MFE of a typical serial link manipulator for assembly operation named PA-10 (Mitsubishi Heavy
Industry, Inc.) are shown in Fig.11DThough the PA-10 consists4805

Maximum jumping height [m]0.4where ki is servo stiffness at the joint i. When the desired
stiffness at hand is shown as the following equation,

0.35
0.3

Fe = S∆x

0.25
0.2

the servo stiffness at each joint is obtained as

0.15

K = J T SJ

0.1
0.05
0

Fig. 10.

0

10

20

30

40
50
φ [deg.]

60

70

80

90

Maximum jumping hight for various leg posturesof 7 joints including a rotational axis around an approach
vector at hand, we omit this terminal joint and simplify it
as a 6 dof manipulator. Total weight of the PA-10 is 32 Kg
and a load mass grasped at hand is a 1Kg cubic-shape object
with the width of 0.1m. Maximum joint torque is determinedaccording to the speci£cation sheet.
MFE

IME

where K = diag(k1 , k2 , · · · , kn ). However, Eq.(28) doesn’t
consider the weight of the load mass nor the manipulator itself.
By taking both weight into consideration, the optimum servo
stiffness for the active stiffness control, which realizes a desired impedance characteristic of the load, can be determinedusing the proposed IME. By substituting Eqs.(8) and (26) into
Eq.(27), the optimum servo stiffness is calculated asτComparison of IME, DME, and MFE for 6-dof manipulator

τ

The IME for various weight of the load mass are illustrated
in Fig.12. The IME becomes small in the case that the load

(30)

= QSJδq + QCJ q̇ − QFbias= Kδq + Dq̇ + τbias

(32)

IME

Hard

Soft
(b) 1Kg

IME(33)

Examples in the cases of high stiffness to the approach andperpendicular directions are shown in Fig.(13).

DME

Fig. 12.

K = QSJ

Mp ẍ + C ẋ + S∆x = 0

IME(c) 2Kg(29)

the following impedance characteristic is obtained.

MFE(a) 0.5Kg= QSJδq − QFbias= Kδq + τbias

In the same manner, by choosing the servo stiffness and theviscosity term as

IME

DME

(28)

Consequently, the load mass acquires the following impedance
characteristic.
Mp ẍ + S∆x = 0
(31)

MFE

DME

Fig. 11.

(27)

Hard

Soft

IME
Fig. 13.

(d) 5Kg
Calculated IME for various load mass

C. The extension to a free ¤ying robot manipulator

mass is light since a majority of produced joint torque isconsumed for the motion of the manipulator itself. On theother hand, the IME becomes large if the load mass gets heavy
since most of the joint torque is transmitted to the load mass.
Active stiffness control[10] proposed in 1980s realizes
impedance behavior of a manipulator hand by utilizing servo
stiffness at each joint. In this method, the joint torque τi atthe joint qi is determined according to a deviation angle asτi = ki δqi

Examples of IME for active stiffness control(26)

The motion of manipulators mounted on a free ¤yingsatellite in outer space causes the motion of the satellite itself
due to the reaction force, since the ground to £x the foundationof the manipulator doesn’t exist [11], [12]. Therefore, themass of the manipulator affects the performance of operationef£ciency more severely comparing with the manipulator on
the ground. In this section, we introduce the extension of the
proposed IME to free ¤ying robot manipulators.
Let’s consider the free ¤ying robot with N joints and N + 1
links. The mass and the tensor of inertia of each link are mi

4806

and Ii , respectively. The linear and angular momentum aroundthe center of gravity of the robot are given as
 



Pg
wI3 0
vg
=
(34)
L
ω
0
I
g

g

g





where w = mi , Ig = Ii + mi r̃gi r̃gi , rgi is a vector fromthe total center of gravity to the center of gravity of the link i,
and r̃ is a skew]symmetric matrix. By differentiating Eq.(34),
we get
 

 


Ṗg
wI3 0
v̇g
0
=
+
(35)
ω̇
0
I
ω ×I ω
L̇g

g

g

g

g

g

Moreover, there is a relationship between the force around thecenter of gravity and the force applied to the hand as follows:
 



fg
I3
0
fh
=
(36)
n
I
n
r̃
g

gh

3

g


=

I3
r̃gh

0
I3

g



Here, H ∗ is the tensor of inertia of the free ¤ying robot in
joint space. By substituting Eq.(43) into Eq.(44), we getτ = (J ∗T + H ∗ J ∗−1 S −1 W −1 )Fh

fh
nh

g

g


(37)

For convenience, we rewrite the above equation as
M V̇g + C = −RhT Fh

(38)

T

where Fh = −(fh , nh ) is force and moment applied fromthe hand to the load.
On the other hand, by using the generalized Jacobian matrix,J ∗ , for a free ¤ying robot proposed by Yoshida et al. [11], thefollowing equation is valid between the velocity of the hand
and the center of gravity, and the angular velocity at each joint.
Vh = J ∗ φ̇ + Rh Vg(39)

By solving Eq.(38) for Fh and substituting the derivative ofEq.(39), we get
Fh = −Rh−T M Rh−1 V̇h + Rh−T M Rh−1 J φ̈J
Mp

Satellite body
1000Kg(40)(41)

From Eqs.(40) and (41), the acceleration of the hand is given
as
V̇h

=

(Rh−T M Rh−1 + W )−1 Rh−T M Rh−1 J ∗ φ̈= SJ ∗ φ̈(42)Thus, the force applied to the load in Eq.(41) can be revisedas(43)
Fh = W SJ ∗ φ̈

→
→

J∗
WS

(46)
(47)

Since S is a function of joint angles, the effect of the load W
at the hand can be converted to the imaginary load W S for
the free ¤ying robot, which changes according to the postureof the free ¤ying robot. For example, let’s consider a task tocatch a satellite in failure using a free ¤ying robot in outerspace. By designing the optimum posture of the manipulator
for catching the satellite in failure, the weight of the satellitein failure W can be changed to SW imaginary. Therefore, it
is possible to reduce the torque required to catch the satellite
and save the energy consumption. Conversely, if we consider atask to release a satellite from the free ¤ying robot to orbit, thesatellite can receive a large releasing force with a small joint
torque by choosing the optimum posture of the manipulator.
To verify the above discussion, we conduct computer simulations to release a child satellite using a 7-dof manipulator
mounted on a free ¤ying satellite as shown in Fig.14. Here,
we assume the size of a free ¤ying robot is 2m×2m×2m, theweight is 1000 Kg, and the length and the weight of each linkare 1m and 10 Kg. The weight of the child satellite is 100 Kg.

Here, we consider a stationary state and ignore the squared
velocity term for simplicity.
In the case that a load with an inertial matrix W is being
held at the hand, the force applied to the load isFh = W V̇h(45)This equation indicates the joint torque which has to be
produced for accelerating the load with the tensor of inertia,
W , with the acceleration V̇h given in Eq.(42). Thus, Eq.(45)
is an extension of the impedance matching of serial link
manipulators de£ned in Eq.(8) to free ¤ying robots.
Comparing Eq.(45) with Eq.(8) in the case that g and q̇ arezero, we can £nd the following correspondences.

hwhere fg = Ṗg and ng = L̇g .
Therefore, from Eqs.(35) and (36), the following equation
can be obtained.


 

wI3 0
v̇g
0
+
0
I
ω ×I ωω̇gwhere S = (Rh−T M Rh−1 + W )−1 Rh−T M Rh−1 .
In general, the motion equation of a free ¤ying robot isgiven asτ = H ∗ φ̈ + C + J ∗T Fh(44)

7-link manipulator
10Kg/link (total 70 Kg)

Payload 100 Kg
Fig. 14.

7-link manipulator on free ¤ying satellite

In the simulation, we examine 2 types of initial postures of
7-dof manipulator as shown in Fig.15, and measure the energy
ef£ciency for releasing the child satellite toward x direction.
The initial posture A indicates that the satellite-side links ofthe manipulator are folded and the initial posture B shows thatthe tip-side links are folded.
The IME for both initial postures are depicted in Fig.16.
From these results, it can be estimated that the initial posture
A has higher torque-force transmission ef£ciency regarding x
direction than the initial posture B in a stationary state.

4807

Z

IME with numerical examples including the selection of an
optimal leg posture for a jump robot, and an extension for
free ¤ying manipulators on satellites in outer space.

XY

ACKNOWLEDGMENT(a) Initial posture A

This research was partly supported by the 21st Century
COE Program ”Reconstruction of Social Infrastructure Related
to Information Science and Electrical Engineering”, and theMinsty of Public Management, Home Affairs, Posts andTelecommunications in Japan under Strategic Information andCommunications R&D Promotion Programme (SCOPE).

ZXY(b) Initial posture B
Fig. 15.
1
0.8

0.8

0.6

0.6

0.4

0.4

0

-0.5

R EFERENCES
1

Fy

0.2

A
-1

Two initial postures

0
-0.2

0.5

0.2

A

Fx

B

1

-1

-0.5

0
0
-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1

-1

Fig. 16.

Fz

B

0.5

Fx

1

Impedance matching ellipsoid

Next, torque-force transmission ef£ciency after accelerating the child satellite with 0.1m/s2 toward x direction is
calculated. Fig.17 shows a ratio of the sum of joint torqueproduced at the joints and the sum of the force applied to thechild satellite which has various weights. The results verify
that the initial posture A has higher torque-force transmission
ef£ciency than the initial posture B even after the manipulator
moved.
0.8|Fe| / |τ|

0.6

A
B

0.4
0.2
0

0

50

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Journal of Robotics Research, vol. 4, no. 2, pp. 3–9, 1985.
[2] R. Koeppe and T. Yoshikawa, “Dynamic manipulability analysis of compliant motion,” in Proceedings of the IEEE/RSJ International Conferenceon Intelligent Robots and Systems ’97, 1997.
[3] S. L. Chiu, “Task compatibility of manioulator postures,” The International Journal of Robotics Research, vol. 7, no. 5, pp. 13–21, 1988.
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100 150 200 250 300 350 400
Weight of child satellite[Kg]Fig. 17.

Ef£cency of Torque-To-Force conversion

IV. CONCLUSIONSIn this paper, we propose a new index for dynamic performance analysis of serial link manipulators named ImpedanceMatching Ellipsoid, or IME.
The proposed IME indicates dynamic torque-force transmission ef£ciency from actuators at joints to a load held at ahand. The IME involves a wide range of concepts proposed so
far as measures of manipulator’s performance. The Dynamic
Manipulability Ellipsoid (DME) and the Manipulating-ForceEllipsoid (MFE) are both derived from the IME as limiting
forms about the load mass. In this paper, we demonstrate the4808