ISMCR2004

A Method to Solve Inverse Kinematics of Redundant Slave Armin The Master-Slave System with Different Degrees of Freedom
Shuhei Nakagawara, Riichiro Tadakuma, Hiroyuki Kajimoto, Naoki Kawakami, and Susumu Tachi
The University of Tokyo, Japan{nkgwr, tadakuma, kaji, kawakami, tachi}@star.t.u-tokyo.ac.jp
Abstract
A master-slave manipulator with different degrees offreedom has some advantages, such as operationalperformance. However, it is difficult to determine theangles of a redundant slave manipulator in such a system.
In this paper, we determine them by measuring theoperator's posture with an acceleration sensor to follow
the operator's motion as exactly as possible.

1. Introduction
There have been extensive research efforts on masterslave robot systems for teleoperation. Since most of them
are for disaster relief and recovery activities, slave robots
are composed of heavy mechanism, preventing the user’ssmooth operation. On the other hand, we have proposed
master-slave system that works in our daily life [1]. In
this system, slave robot can not only perform precise
handiworks, but also communicate with surrounding
people with gestures. We call this concept “mutualtelexistence”, because the operator virtually exists in theremote place not only from the operator’s view, but also
from surrounding people.
The specification to satisfy these two demands (precisehandiwork + communication with gestures) is as follows.
First, the slave arm must have 7-DOF in order to express
an operator’s gesture. Just like human arm, the first andlast three joints must intersect at single points, which areshoulder and wrist, while the remaining one joint
represents elbow (Fig.1).

The second specification is master arm. When the slave
arm has redundant degree of freedom, common bilateralcontrol methods are the symmetric position servo type
and the force reflecting servo type between corresponding
joints by using exoskeleton type 7-DOF master arm.
However, to restrain an operator with active mastermechanism is not only dangerous but also leads torestriction of an operator’s motion, which makes therange of operation narrower because of the mechanism
complicated by components such as motor and sensor.
Therefore, as for master arm, it is desirable to have 6DOF, not 7-DOF, and perform force feedback control
adequately without restraining an operator by mechanism
except for his wrist. In our prototype master manipulator,
the last three joints intersect at a single point using gimbal
ring structure (Fig.2).

Fig.1 Configuration of the slave arm

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2. Master-slave manipulator with different
degrees of freedom and its inverse kinematics

Fig.3 Concept of master-slave system with different
degrees of freedom
As the operator’s arm can be regarded as a redundant
manipulator, conventional methods to solve inverse
kinematics of redundant arm can be applied to measure
the operator’s posture. One of the popular methods is todefine a swivel angle of an arc of a circle, which theelbow traces, lying on a plane whose normal is parallel tothe wrist-to-shoulder axis [4]. Given the wrist position,orientation, and elbow swivel angle, the algorithm cancompute the joint angles analytically. Note that the wrist
position and orientation of the operator is identical to thatof the master manipulator. In order to get the remaining
information (swivel angle), we used the acceleration
sensor (ADXL202E, Analog Devices, Inc.) attached on
the operator’s upper arm. The sensor has enough highfrequency response (1kHz), while its small size andlightness in weight allow the operator to move freely. We
used an axis of the sensor corresponding to the changeof the swivel angle (Fig.4).

As our master arm has 6-DOF and slave arm has 7DOF, simple symmetric servo between corresponding
joints cannot be applied. Remember our two demands:
precise handiwork and communication with gesture.
From the first requirement, the position and orientation ofslave arm’s wrist must coincide with master’s. 6-DOF isused for this purpose. From the second requirement,
slave’s posture must match as close as possible as
operator. Remaining 1-DOF is used for this purpose. Notethat the conventional method based on the pseudo inverseof Jacobian matrix is not appropriate, because it does notsatisfy the second requirement.
There are some possible ways to measure an operator’s
posture such as using markers and a camera, which isadopted in general motion capture system [3]. However,
the optical method has two problems: time-delay andocclusion. Therefore, we take another simpler method
mentioned below to avoid these problems. By thismeasurement, master side has 7-DOF for position andorientation, while it has 6-DOF for force feedback(Fig.3).

Fig.4 Swivel angle , an acceleration sensor attached
on an operator’s upper arm, and its axis aFig.2 Appearance (top), configuration (middle) of themaster arm, and its gimbal ring structure (bottom)

The method to solve inverse kinematics of slave arm is
as follows. The origin (0, 0, 0) is coincident with theshoulder position and L1 , L2 , L3 are the length of theupper arm, lower arm, and the distance from the goal
position to the shoulder of the slave arm. First, the goal
position P and orientation R (rotation matrix) of slave
arm’s wrist are determined by forward kinematics ofmaster arm. Then the joint angle of the elbow θ 4 isdetermined geometrically (Fig.5)

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the wrist, three joint angles of the wrist θ 5 ~ θ 7 are

L12 + L22 − L23cos θ 4 =.
2 L1 L2

determined.
As described above, we can solve inverse kinematicsof the redundant slave arm which satisfies the wrist’s
position and orientation obtained by the master arm andelbow position computed from an acceleration sensor’soutput.

3. Experiment3.1. Procedure

Fig.5 Relation between shoulder, elbow, and wrist
In Fig.5, n is a unit vector from the shoulder to thewrist. Additionally, u and v re unit vectors which form
a local coordinate system for the plane containing thecircle which elbow traces. u is determined arbitrarily tocorrespond to φ = 0 . Then v is set v = n × u .
Similarly to θ 4 , α is determined and the center of thecircle c which elbow traces and its radius r can becomputed ascos α =

L +L −L,
2 L1 L3212
322c = L1 cos αn , r = L1 sin α .
If we set u the direction that the output of theacceleration sensor a is equal to the gravitational
acceleration gu=(n z0 − nx ) Tn 2x + n 2z,

the swivel angle φ is obtained assin φ =a.
gThen the elbow position E(φ ) is given byE(φ ) = c + r (cos φu + sin φv).
From the relation between this elbow position E(φ ) andwrist position P , three joints angles of the shoulder
θ1 ~ θ 3 are determined. Next, from the orientation R ofWe constructed a master-slave system consisted of 7DOF slave arm, 6-DOF master arm, and an acceleration
sensor attached on an operator’s upper arm. We examined
the accuracy of slave arm’s elbow position by comparing
it with that of the operator. The elbow position of theoperator is measured by a link type measurement
instrument (ADL-1, Shooting Star Technology). The joint
angles of the slave arm are determined by the acceleration
sensor using the above algorithm. In the first half period(0∼ 30 seconds), the operator does not move except for
his elbow and in the latter half (30∼ 60 seconds), hemoved his arm freely.

3.2. Result
The result is shown in Fig.6. In the first half, the gap of
the elbow position between an operator and slave arm iswithin 5[cm], while in the second half, it is within 10[cm].
At the same time, we observe quite little time delay. The
mismatch of the elbow position is caused by thefollowing two reasons. One is the difference of arm
length between the operator and slave arm. The other isthe wrong value of the acceleration sensor. We assumed
that the sensor measures gravity direction, but it alsomeasures the acceleration of the arm itself. The easiest
way to solve the latter problem is to apply low-pass filter,but we must be careful about the trade-off between
strength of the filter and time-delay. Another solution isto use a gyro sensor together with the acceleration sensor
to compensate for the acceleration of an operator’smotion. However, as our purpose of measuring the elbow
position is for communication with gesture, the accuracy
got by only an acceleration sensor is enough.

4. Conclusion
In this paper, we introduced a method to determine thejoint angles of redundant slave arm by the measurement

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of an operator’s posture. Especially we examined the casein which master arm has 6-DOF and slave arm has 7DOF. As our purpose includes communication withgesture, the conventional method based on pseudo inverseof Jacobian matrix was not adequate. We used anacceleration sensor attached on the operator’s upper arm
to measure redundant DOF. By experiment, we showedenough accuracy of the slave elbow position computedfrom the output of the acceleration sensor.

Fig.6 Elbow position of x, y, and z-axis

5. References[1]R.Tadakuma,
Y.Asahara,I.Kawabuchi,H.Kajimoto,N.Kawakami, and S.Tachi: “Development of an Exoskeletontype 6-DOF Master Arm for Telexistence,” International
Symposium on Robotics (ISR2004), TU33-3, 2004.
[2]Y.Inoue, S.Tachi, and H.Arai: “A Method of ImpedanceControl of a Master-slave Manipulator System,” Journal of theRobotics Society of Japan, Vol.10, No.4, pp.480-488, 1992
(Japanese).
[3]H.Hasunuma, H.Kagaya, M.Takatori, J.Fujimori, F.Mifune,S.Shikoda, M.Kobayashi, T.Itoko, and S.Tachi: “Developmentof Teleoperation Master System with Kinesthetic Sensation ofPresence,” Proc. International Conference on Artificial Realityand Telexistence (ICAT1999), pp.53-59, 1999.
[4]D.Tolani, T.Goswami, and I.Badler: “Real-time Inverse
Kinematics Techniques for Anthromorphic Limbs,” GraphicalModels,Vol.62,
pp.353-388,
200

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