tEEI, l RANSACTIONS ON ROROBTI〔S A:',D ALTOMATtO,;, VOL 9. 1\0 I. Fl:B且UARY 1993。；「＼ -,
＇

.'?'

』

C
ヒω
セω
一
己院内守

[14] D. Y. Kim. J. J. Kim. P. Me町、D Mint,. and A. Rosenfeld, "Robust
computer vision: A least-median of squares based approach，＇、in PruζDARPA {magどじnden1anding Wrnkshοp. 1989.
[I丹J S.S. Sinha and R. G. Schunck, "A two stage algorithm for di只continuity
preserving surface reconstruction, Jf:;Jo'E Trans. Pattern Anal. Maζ·hine
Imel/.
1 14, pp. 36-55，』an. 199ユ
[16] A. E. Beaton and J. W. Tukcy, "'The自tting of power 田町人meaning
polynomials, illustrated on band-spectroscopic data，刊 Technometrics,
vol. l弘pp. 147-185, May 1974.
l 17] D. Andrei\ s, P. Bickel. F. Hampel 司 P Huber, W. Rogers, and J
Tukcy, Rρhu<t E叩mates ρif Iρcation: Survey and Advances. Princeton
Princeton University Press, 1972.
118] P. W. Holland and RιE Wei叫h’rcwcightcd lea尽T squares‘，司CθFηm“，，. Stθrist. Theor. Mcrh., vol. A6, no9. pp. 8Ll一827’1977.
[ 191 C Goodall, ..M estimator of location: An outline of the theory，”mUnderstanding Rob削f and F.xplorato 円 Dota Analysis, D. C. Hoaglin,
F Mosteller. and J. W. Tukcy、Ed只 New York: 1983, pp . .139-403

町 、。

／

lb)

(a)

85

(C)

Fig. 11. The percent rclati可C error in (al the Gau只，ian cu円aturc and け1J themean cu刊ature as a function of tuning constalll for the りnthetic pate h. S IIJ
curve: dean. Dashed curve: with normal noise. Dotted curve: m》rmal and an
outlier. Dot-dash cu円e with outliers. (c) Real spherical patch。

Dynamic Control of a Manipulator with
Passive Joint� in Operational Space
Hirohiko Arai. Kawo Tanie. and Susumu Taιhi

1hυ

）
（

Abstract-We present a method to control a manipulator with p回si刊
jomts. which have no actuators, in operational space. The equation of
motion is d也、cribcd in terms of uperational coordinates. The coordinates
are separated into active and pa.mve componen臼. The acceleration of
the active components can be arbitrarily adjusted by using the coupling
characteristics of manip凶ator dynamic,. This method is also extended
to path tracking control of a manipulator with pa，明ive joints. A desired
path is geometrically sp町i詰ed in operational space. The position of themanipulator is controlled to follow the path. In this method. a pathcoordinate system based on the path is defined in operational space.
The path coordinates consist of a component parallel to the path andcomponen臼 normal to the path. The acceleration of the components
normal to the path is controlled according to feedhack b出ed on tracking
error hy usin里the dynamic couplinJ! among the component�. This in torn
keeps the manipulator on the path. The effectiveness of the method is刊rifled by experiment,、山ing a twn-degr町－of-freedom manipulator witha passive jmnt.

(aJl. ]NTRODUCTIO:-l
The number of degrees of fr巴edom of a conventional manipulator 1s(c I

equal to the number of joint actu品tors. Since the mass of the actuator

id)Fig. 12. Real range image of a jumble of packages. (a) Raw _ data mcludmg
noise and outliers. ReconstrU叫ed b1quadratic surface, to ma,imal rcctan匹ula,
footprints. (b) Cylinder. le) Rectanguler hox. (d) All objectsof aぉerial type manipulator is a load for the next actuator. the sizeof the actuatorぉhould increase rapidly from the wrist joint to thebase joint. As a result‘the base joint must be equipped with a hugeactuator compared lo the load of the manipulator. In order to decrease[9] R. M Harahck，“Computer \ i sion theory: T he lack thじrcof, 仁川npur
Vision, Grnphiιs, Imoxe Process.
I. 36. pp. 372-386 1986.
“[10] W. F,mtnc仁 Rcli泊bility analy則。f parameter e,timal!On in l印刷了
models with application to mensuration problem, in computer i,ion 可Comput. Vision, Graphics, Image Process., ,ol. 40, pp. 273 3 lO、1987.
[11] P. J. Besl, J. B. Birch, and L. T. Watson，“Robust window operators，刊’
in Proc. Int. Conj. Comput. Vision ( /CCV 88人1988.
[12] R. M. Haralick, H. Joo, C. Lee, X. Zhuang, and V. V. M. Kim,
“Po田estimation from corresponding point data,' IEEE Trans Svst. Man
Cybern., vol. 19, pp. 1426-1446, 1989.
[13] D. S. Chen，“A data-dnven mtermediate level feature extraction algorithm," IEEE Trans. Pattern Anal. Mα chine Intel/., vol. 11, no. 7, pp749 758, 1988.

司、。

、

刊

the I\ eight, co,t, and energyじon,umption of a manipulator. various
methods have h巴en proposed for controlling a manipulator that has
more degrees of freedom than actuators [ 11. However, these methods
Manuscript received August 8, 1991; rcvi唱d April 17, 1992. A pa円mnof this work was presented at出e 1991 IEEE International Conference on
Robotics胡d Automation, Sacramento, CA, April 9- 11, 1991，四d at the
199 l IEEE/RSJ International Workshop on Intelligent Robots and Systems,
Osaka, Japan, November 2-4, 1991.
H. Arai and K. Tanie are with the Mechanical Engineering Laboratory,
AIST, MIT!, 1-2 Namiki, Tsukuba, Ibaraki 305, Japan.
S. Tachi is with RCAST, The Univer百ty of Tokyo, 4-6-l Komaba, Meguro
ku, Tokyo 153, Japan.
IEEE Log Number 9205078

ゆ42・296X/93$03.00 © 1993 IEEE

一一一一一一一一｝

86

E彊Eτ'RANSI\CTIONS ON ROBOBTICS /\ND I\UτOMI\TION, VOL. 9, NO. I, FEBRUARY 1993”’

化sired acceleration is generated at theぉtive components, which
are equal in number to the active joints, by using dynamic coupling
among由e comp<inen胞．百is method is ex飽nded to pa血・住民king
con位。，l of a manipulator with p路島ive joints. A desired p血is
geometrically spc割cified in operational space. The tip position of themanipulator is controlled to follow臨desired pa曲. In Section m,
戸zth coordinates are defined 凶 a kind of operational coordina包sys飽m based on 血e des出d p抽. The p飢h coordin蹴s consist of
a component parallel to由E desired pa白arid components normal to
出e desir官d p柚. In Section IV, a method for path－回eking control is
described. The acceleration of曲e components normal to the desired
p抽is con位oiled by using the dynamic coupling among components.
百1is in turn keeps the manipulat町on曲e desired path. In Section
V, the effectiveness of恥仰が蹴d methods is demonstrated byexperi111Cnts using a 2-DOF ma国凹1成。r w抽a passive joint.
II. CONτ官OLIN OPERAτ10NAL SPACE
A. Equation of Motion with Operational Coordinates）
1（

Ann-OOF mar岨伊Jlator is consi<命誌ed here.百1e operational space isassumed to be n-dimensional. We assume r DOF of the manipulator
are active joints.百e remaining 叫 － r DOF 師 passive joints withholding防法.es inst伺d of actuators .
胃腸eq1凶.tion of motion of由e mar岨.pulator in joint s抑m can bewntten as
M(fJ)8 + b(fJ. 9) ＝瞳
where
6(9,9) = h(9,9) + I'9 +g(9)

。ー

and

εIll”
joint 叩.gle vector,
εRnjoint to岡田vector,ngra吋ty tor司ue v配tor,g(O）εRnCoriolis 組d cen凶fugal回que vector,
l&(fJ, 9 ）ε1R
nxn
町出a m甜ix,
M(IJ ） εnn
I' E ll xn
viscous制ction ma凶X.

百ie el，自問：nts of• are 問arranged as
旬口

［r

T

r

(2)

o] .

Tεm. is the torque of血e active jo泊ts, and the torque of 血e
p蹴.ivejoints is zero. Accordingly, M and 6 are also rearranged andpar甘tioned as follows:r

ap『EBEESE－td

FBEE－EEEEL

[!:]•

EO EO

M=

一
一
置。

requ甘'C s;戸cial mechanisms in additi叩 to basic links and joints. In
曲is paper, a 四時曲叫 is関節目制for con回Uing a mani伊1lator蜘I
has more joints命制 actuators without using a制itionaJ II即：hanisms
except joint brakes.
百1e d wnics of a manipulator have nonlinear and co臼pl泊gcharac蜘istics.明1en鋪ch joint is co附oil叫by a local Jin伺r
feedback loop，血.ese factors result in dis加加問s.百e eliminationof such dyn創凶c disturbances hぉ民側one of 白E major problems
踊mani ulator con1回I (21イ4). A design由民町for a manipulator.
with nei出er nonlinearities nor dynamic coupling has also bi沼npro戸>Sed [5]. However, the effects of these dis伽rbances are available
to命ive a joint 白紙 in itself does not have an actuator. Suchdynar泊c ch釘ac飽ristics are actively used in human handling tasks.
For example, when a heavy load is handled, all the muscles of白e加m如 arm are n脱出自ce図鑑ily u関d. Some joints, e.g・， wrist joints,
are kept free，制d the inenia of 恥load is utilized effectively. Sucha dynamic skill will also be signi自cant for robot con回I. Some robot
control schemes that u且lize dynamic coupling effects have previously
以潟n prop<鴻ed (6), [7].
As a means of co曲。Uing a manipulator with 即時 joints thanactuators with叩t using additional mech姐isms, we pro伊，se con回1ling passive joints by 岡田g dynamic coupl同（8]. We develoi削飢algorithm fぽ凹凶－to-伊血t con位。l of血e m副pula伽飢d柳，lied it
to a two-degree-of.・貨切dom(2心OF) mani伊Jlator [9]. In也is method,a manipulator is com戸鴻ed of two ty戸S ofjoin随：ac伽e創1d開問ve.
Each 配tivejoint consists of 組 actuator 組d a position sensor (e.g・－
an encoder). Each passive joint consists of a holding肱ake and apositio目印n 関，r. When the brakes of the抑制ve join箇制官 engaged,
白E創刊join臼can be con回lied wi血out affec出g山由旬of thepassive joints. When the防法.es 蹴 released，泊施passive joints can
削減e合eely. The mo1品.on of the active join悩generat白紙celera勘nof 血E P制ive join包via臨coupling charac削剥cs of manipul蜘rdynar凶.cs.百lC抑制vejoin胞can be controlled indirectly回出is m佃・n町. The to凶position of the manipulator is αmtrolled by combining
these two c棚，trol m叫es. Jain and R叫riguez indepen伽itlypro伊副
a similar t釘hnique to control a m副pulator with passive hinges.百ey
also devel句制組 efficient dynamics algorithm 'using s戸tial operator
algebra [ IO].
W hen some of the joint ac制ators of a manipulator are exchanged
for holdin�伽司kes with白is meth叫，we c組 build a lightweight,
energy-saving, low倫cost manipulator. We can take advantage of
由ese m剖包by apply泊g世記me血od to simple assembly ro加，ts,
con回1 of r叫undant mani仰lators, etc. Space applications (e.g.,space mani凹Jato隠，expansion of space struc加re) may be feasible.
百is me出od c組 also be applied to failure recovery con回，1 of amanipulator (11).
C抽出I with 恥 p蹴ive joints relea剖is an e悶ntial p蹴of
血is me曲叫. In [8] and (9], we controlled the manipulator 泊 joint
space. In血is approach, a desir吋回�ectory is assigned to白e passivejoin飴，組d the motion and剛司ue of批activejoints is calcula包d torealize the desired motion of the passive joints. The motion of出e
active joints . is determined by 曲e des祉吋佐官？ectory of the passive
joints and the dynamic coupling among the joints. Consequently, themotion of the tip of the m組ipulator cannot be prescribed. However,the position of the tip in operational space, e.g., Cartesian space, isusually important for practical manipulator tasks. It is necess 訂y tocontrol the path along which the tip moves if the manipulator is to
avoid collision with an obstacle.
In this paper, we extend our previous work on control injoint space
to control in operational space. In Section II, the control scheme inoperational space is presented. The equation of motion is represented
in terms of operational coordinates. The operational coordinates
are separated into active components and passive components. The(3)nwhere Ma E 1R: x0 ,Mp εm,(n-r) x n,b a εR", and bp εR -r.
The equation of motion of the manipulator is rewritten in termsof o戸田rational coordina回p E JR". We assume白紙曲e operational
coorせina蜘組d the joint coordinates are related as follows:

p=J(/

(4)

where J(B ） ζJR" " is a Jacobian ma仕ix. When(4) is differentiatedwith respect to time, we obtainxp= Jij +je

(5)

If J is nonsingular＇

ii=r （長一jil).

(6)

Note that the manipulator has n DOF and is nonredundant. Whenthe manipulator is redundant, matrix J is not invertible. One way toIEEE TRANSACTIONS ON ROBOBTICS AND AUTOMATION. VOL. 9. NO. I. FEBRUARY 1993

inve口J is the extended Jacobian method [12], in which auxiliary
coordinates 訂e added to the operational coordinates so that J canbe inverted.
The r components of the operational coordinates 司 which should becontrolled, are chosen and grouped as z. The remaining components
are grouped 副首・ He陀， F and E 三 J- 1 are rearranged andpartitioned as follows:

E

E

H

レ－

一
一

rEa－－EEE・L

z g

旬r

一
一

{7)

where zεR ’ Jεffi" '.H a εR nx,·. and H p E ffi"X(n r)
We define :,; as an active component and y as a passive component.
The desired motion is assigned to the active components while thepassive components are controlled so as to realize the desired motionof the active components. When (3), (6), and (7) are substituted in(I), we obtain
MαHaii: + M0H/y-M,,HjiJ + b,, = r

俗的Mp Haii: + Mp H r ii-MrHjiJ + br = 0.

(8b)The equation of motion is represented in terms of the active andpassive operational coordinates. Moreover, it is divided into (8a）‘which is related to torque of the active joints, and (8b), which isrelated to torque of the passive joints.

B. Control in Operational SpaceThe desired acceleration can be generated arbitrarily at components
equal in number to the active joints. Control of the active components
is given priority, and the computed torque method (21 is applied.
It prescribes a desired tr司ectory for the active components andgenerates the motion of the passive components in order to realize
the desired 町民jectory of the active components.
The desired position Zd, velocity :Cd, and acceleration id of theactive components are obtained from the desired trajectory. The
following PID control is applied to suppress the tracking e汀or:
品＝ね＋ Kv(:i:d -:i:）＋ι （Xd-:i:）＋広三 J(xd 一川 θ）

where Kv . K,, . and K；εR ,. x ' are the diagonal gain matrices.
When measured values of the joint angle and velocity are substi­
tuted in /J and /J of (8a) and (8b), each component of M. H. J‘andb is determined. If品of (9) is assigned to the acceleration ii; of theactive component z, (8b) can be considered as a linear equation with
reg訂d to量. If Mp H p εrn < " r）叫ト叶is nonsingular (and hence
inve目ible), (8b) can be solved uniquely asii=(MpH,,) 1(-M,,H 0 品＋M,, Hjii-b,,).

(IO)

The nonsingularities of J and M,,H,, are essential for this method
The singularity of M,,H,, (dynamic singularity) will be discussed
later.
When ( 10) is substituted in (8a）『 the torque r to realize theacceleration晶J can be determined.
r= {M,, -M 0 Hp (M,, H,,l-'Mv }(H,, x:i-ajil)+bα－M,,H,,(M,,Hp ）

ー＇

b， ，・

（II)

When we apply this torque T to the active joints 司 we will obtain theacceleration云:i. The active components are guaranteed to converge
to the desired values if Kv . K,, . and K; are chosen such that all thepoles of出e feedback system are located in the left half plane. Fig. I
represents a block diagram of the control system.

87

Xd

Fig. I.

Control system in operational space.

C. Function of Brakes
The brakes of the passive joints are used to set up the initialconditions. When the brakes are engaged, the manipulator has r
degrees of f問edom The passive joints are fixed, and the angularvelocity is zero. The number of the active components is r. The
active components are rep，官sented by a kinematic equation including
the angle of the active joints. The陀fore, the initial angle and angul紅
velocity of the active joints that realize initial position zo and initial
velocity :co of the active components can be obt幻自吋by inverse
kinematics.
The initial conditions of the passive components cannot be set
up by the active joints alone. If the initial position of the passive
components needs to be set up, it is necessary to use the point-to
point algorithm of [8]. The initial velocity of the passive components
is determind by the initial velocity of the active components.
The自nal conditions of the passive components are determinedby the control hysteresis of the active components. Even if themanipulator is at rest in the initial condition and the final velocityof the active components is controlled at zero, the final velocity ofthe passive components is generally not zero. In the experiments, weforced the passive joints to stop with the brake. In this method, thepassive joints cannot stop at an exact angle because of the time delayin the brake operation. It is necessary to switch仕om operational
space con甘ol to joint space control and to stop the passive joints
befo陀 braking if the purpose of the control is positioning.
目

D. Dy�凶 mic Singularity
The conditions required for this control method are as follows:I) Matrix H can be obtained. (Jacobian matrix J is invertible.)2) Equation (8b) has a unique solution. （量is determined uniquely.)3) In (II)、r and ii:」 show one-to-one co 町espondence.
Condition I means that the manipulator must be kinematicallynonsingular. Condition 2 is equivalent to ti】 E nor】 singularity of matrix

Mmatrix {Mσ 一 MαH,,(MrH,,) 'Mµ }H，ヨ. Conditions 2 飢d 3 a陀mathematically equivalent. The acceleration of the passive compo 一
nents cannot influence the acceleration of the active components
in the positions where MpHP is singular. The acceleration of theactive components is determined by the position and velocity of themanipulator ‘児gardless of the acceleration of the passive components.
i
Therefore, th 日method is difficult to use near these dynamic singularpoints.
The condition of dynamic singularity （心t(Mp Hp ] = 0) provides
one degree of constraint For example, when a m 佃ipulator has 3
DOF, the dynamic singular points compose a surface. The manipuJator cannot pass through this surface in the proposed method. An
algorithm to solve this problem is necessary. One way to solve thisproblem is to use the brakes and fix the passive joints. Anothersolution is to switch the choice of the active components. The inertia
matrix M is generally nonsingular. Therefore, the rank of matrix
M P is 11 - r. Singularity of MrH" depends on H v· If the active
components are chosen in a di仔erent way, the manipulator can passnear the dynamic singular points. (Of course, control of the former
active components is compromi 民d in this U田．）
目IEEE TRANSACTIONS ON ROBOBTICS AND AUTOMATION, VOL 9, NO. I, FEBRUARY 199388

in terms of path coordinates. Note白紙there exist lots of pathcoordinate systems for one desired path. I白血e case of出1s example.
a spherical coordinate血at has its origin at the center ofthe desin吋path c組 also be a path coordinate.

,x,

二ロで
/

x,

Desir出Path(n;3)

IV. PAIB-TRACKING CONTROL

Fig. 2. Path coordinate system

In this section, th巴 path－甘acking control scheme is described. The
control scheme is based on由e pa白 coordinates. First, the equationof motion is represented in terms of the path coordinates. Thenthe path tracking control scheme with feedback is proposed. The
initial conditions ofthe desired path are discussed. Path甘ack:ing withmultiple passive joints is also considered.

III. PA百I COORDINATESIn the following sections, a method for pa血tracking control
ts presen匂d A mathematical description of the desired path isconsidered at first. The desired path is geometrically specified asa continuous curve in operational space. It is not associated m白a time variable. In minimum-time tr司ectory planning problems, this A. Path-Tracking Control
type ofpa出 is often p紅ameterized by a path parameter [13]-[15］.百1eBoth operational space and the path coordinate space are assumed
position ofa point on 出e path is 陀presented as a vector function of
to be n-dimensional. It is assumed that the manipulator also has na scalar p 紅ameter. When operational space is n-dimensional, a point
DOF, and it consists of n - 1 active joints and one passive joint.
qεlR.n on the path is represented as
We propose a con紅ol me血od in which the manipulator tracks theq = q(s).so壬s 竺 SJ
( 12) desired path defined in Section III. In Section II, it was shown thatthe components句ual in number to the active joints can be active
where s目白e pa出 parameter. q(so) is the start point of the pathcomponents and con町ollable. The number of active joints of theand q（り） is the end point. s can be considered as a distance alongpresent manipulator is n - 1. The number of components of thethe pa出. Since s is a scalar, this me血od can represent a point only
path coordinates is n, and the number of components x normal toon the pa出 itself.
the path is n - 1. Therefore, x is田ated as an active component andReal-time path tracking control is considered here. If the manip­s is treated as a passive component.
ulator deviates from the path due to disturbances, feedback control
From (16), the operational coordinate q and the path coordinate pshould force the manipulator to印刷m to由e pa血Therefore, points
are related as q = q(p). The operational coordinate q is calculated
not on the pa廿1 as well as points on the path should be represented.
by forward kinematics, q = q(B). When
Furthermore，出e住acking error should be measured. We propose theconcept of path coordinates as an extension of the path p紅創neter.
θq
δq
’
Ji =
I
、
·
J1 、J2 E IR
A curvilin伺I coordinate frar即時defined in operational space. TheDHθpcoordinates are composed of a component s along the path andcomponents z ト
X n I normal to s. .r; is normal to x 1 if i =I j. the Jacobian matrix J (/J) E JR" x " of path coordinate p for joint
百1ese coordinates are called path coordinates (Fig. 2). A point coordinate IJ is represented as
p E JR" represented in terms of出e path coordinat酎 is
J＝ 坐 ＝J-;
' 'Jト
T
θ/J
p= [xぃ· ·,X n 1,s] = [.:i? s] T
(13)
Consequently, p 組d B are related as
百1e desired path is represented as
合＝JiJ.
(17)
so 壬s ざ SJ
(14)
x = Xd (constant),＝

・司

m terms ofthe path coordinates. The desired path is also representedasq = q([xd

T

T

s] ),

so壬s壬SJ(15)

m terms ofthe operational coordinates. Equation (15 ）ιan be extended
to rep陀sent all points in operational space. A point q in operational
space is represented as a vector function of path coordinate p.
q =q(p).

Path coordinates: Cylindrical c副賞dinatesq(p) = q((x1, x2, sf)= (x1 cos(ws) + xo x1 sin(ws) + Yo,.i:2],
司

明1e desired path is represented as:,; = [x1,x2f = [ro, .:of

M(B)ii+b（仇0) = u.

(18)

The equation of motion of the manipulator is rewritten in terms ofpεJR". The joint torque vector u is composed of the active joint
torque, rεIR" 1, and the passive joint torque(= 0).
包＝ [r

(16)

巳quation (16) represents the coordinate transformation from the pathcoordinate space to operational space.
Example: Operational coordinates: Cartesian coordinates （η＝3).
Desired path: A circle of radius r0, ce附red at [:ro, Yo 『 ご口］， parallel
to白E xy planeq(s) = (ro cos(ws) + xo, ro 司叫ws) + Yo, zo],

The equation of motion in joint space is written as

r

OJ

I

(19)

Then the same procedure as in Section II results in the following
equation:
Ma H a x十 Mα H,,s-Mα njiJ+bα
where
M=

T

(20a)

MpH a :i:+MpHps-MpHjiJ+bp= O(20b)[!!:]·

b=

[!:].

1

＝

r = H = [H a

Hp］

ο

n
d M.εJR(n-l)xn,MpεJR lxn ,ba ε IR -1,bpεIR'.H. ε]Rnx (n I), artd HpεJR" Xl.
In o吋er to 主田p the manipulator o目白e desired pa由，出e :,:componen飴normal to the path should rem細目 constant at Zd・The
組

IEEE TRANSACTIONS ON ROBOBT!CS AND AUTOMATION. VOL. 9. NO. I. FEBRUARY 1993

F,g. 3Path-trackmg control system司

acceleration是J is generated to adjust x to Xd・The following PID
control is applied:
品工－Kv x +K山 － x)+K,j（山 一 刻dt

(22)

，

where Kv, Kv, and K εm(u-[)X(u-1) ar巴the diagonal gam
matrices. x and £訂E the measured values of the position and velocityof the x components.
最 J calculated from (22) is substituted in 云of (20a) and (20b ).
If MpHP is not equal to zero, the torque T to realize x/,, can be
determined as
r= {Mα － MaHp (Mv H µ ) 1Mµ }(H a x』

HJf!)

tr勾ectory of the manipulator is obtained. The final value of s 1 andthe time period to travel along the path 釘E also evaluated.
If the initial velocity along the path is too slow, the direction of s
may be inverted before reaching the final point, and the manipulator
cannot complete path tracking. Therefore, it is desirable to estimate
the minimum initial velocity necessaηto reach the白nal point. M,
and HP in (24) can be represen旬d as functions of s only. On thepath，。＝ H,,.,. When the friction of the passive joint is negligibly
small and I,,, is composed of Coriolis, centrifugal, and gravitational
forces, (24) can be rewritten as
ぷ＝ f(;;)i + y(s)?

(25)

where f(.s) and g(.s) are functions of sand do not depend on initial
velocity ・.so. Note that
d.,

ィ／，
dt

,l.s

1
2

d（.�） 2

(26)

ti.,

Equation (25) is a linear differential equation with respect to i ？ . 明1e.?(.,) ＝《＇Xj)(23)

十ム－ M,.Hp (M,,H,,) 1b0.

89

(.£:

2 川 正）［バ＋

When we apply torque T of (23) to the active joints司 we will obtain the
x components acceleration of (22). The deviation of the manipulator
from the desired path converges to zero asymptotically if Kv , Kv.
and K; are chosen appropriately. As a result, the manipulator tracks
the desired path
Fig. 3 shows the path tracking control system.

j' 2g（川） ( { 2f（ぞ）dぞ）

d(].

(27)

I

l'X］》（ ＞川正）＞｛）目

悶

B. Planning of Desired Path and Initial Conditions'<G > -As an initial condition, it is assumed that the manipulator is moving
with sufficient initial velocity in the direction of the path when
path tracking control begins. This can be achieved, for example, by
accelerating the active joints while the passive joint is fixed and thencommencing the path－仕acking control with the passive joint released.
The desired path should be planned geometrically so that the initialconditions can be realized by the active joints only. The manipulatorhas 11 - 1 DOF when the passive joint is自xed. The initial a川gularvelocity of the passive joint is zero. The initial position and directionof the desired path 訂E limited by these conditions. On the other hand,
the initial velocity along the path can be controlled arbitrarily by theact! ve JOmts.
When path tracking is performed as part of point to-point control,
the passive joint should stop at an exact angle. The final directionof the path is also limited so that the angular velocity of the passive
joint is zero.
C. Velocity along PathIn this method, the .s component is accelerated/decelerated in orderto realize the desired value of the x components. Therefore, the scomponent cannot be controlled directly. The time period necessar下F
to travel along the path depend 討 on the じ onligurはlion of the path andthe initial vcloじity along the path
From (14）、z is constant when the manipulator mo C 丸 along thedesired path. Thus‘ x = 0 and云＝ 0 on the path. Since Hj
iIJ.
acceleration along the pはth is

、

二

d

二

一

（M1,H1,, 1(M1 ,H 1，、十 I,,.)

一(24)from (20h). The initial value リi 乃山内し， , and the final valuじ IS -' t
When initial values -'u and 内arc gi en ， 何 and ., can he calculated
by numerical integration of ( 24) for 1, 5対壬.， 1 Thus ‘ the time

、

釘

L

2,<J（正）ぽp（ーか川ぞ｝1(.

(28)

The right side of (28) is calculated by numerical integration. If ,so
is chosen so that .,j is larger than the maximum of the integration
value, the manipulator can reach the品nal point. It is clear from (28)that if y （ 川三0 for町：：：： ,; :::: .s I, the manipulator can reach thefinal point for any .,u > 0.
The initial velocity necessary to reach the desi陀d velocity at the
final point or at a point halfway along the path can be calculated if
the initial velocity is in excess of the minimum velocity. When (24)is integrated backward for so :S巧三川with δ and ii given final
values -'d and川，the initial velocity .,0 to 陀alize the desired velocity·'cl is derived.
During real-time control,
.S
·'U
、
ハ
(29)
KO
与 f
ー

and （）くλく1 on the path. The value ofλis monitored while themanipulator is tracking the path. The control terminates when λ＞ 1.
D. Control of a Manipulator and Multiple Passive Joints

A manipulator composed of 11 ー 1 active joints and one passive
j川nt is considered in Secti，川、IV-A to IV-C. However. the propo 柑dcontrol method i 只 ea日ly extended to a manipulator with multiple
passi E joints. In 11-dimcnsional operational space、it is necessary to
じ ontrol 11 - lιomponents to keep the m 出1ipulator on the desired
path. Therefore. u ー 1 activ巴 j川nt, are neじ出 sary. The manipulator i 討
品川1 mcd to have 111 > 1 passive joints. In this case, the manipulator
is redundant in operational space. and the extended Jacobian methodin path Cl附rdrnはte space is 辻pplied. Tile equation of motion is、

M1H ） が＋ b(4.印）二u

u = [,

Ol

T

(30)
(31)

鹿島E TRANSAC官ONS ON ROBOB＇百CS AND AUτOMATION, VOL. 9, NO. I, FEBRUARY 1993

90

TABLE IPARAMETERS OF Tl到E MANI!町LAIUR

mI

Encoder

h
OU

MW

0.5

1

・

fin皿［s]

1.0

Ti皿I凶

1.0

(a)

Model of 世田m副首.pulator.

【圃】Fig. 5.

】

1\vo-degr田-of-fr田dom m加ipul蹴町．

お

2.0 kg
1.0 kg03 m
2.2 N·m-s/rad
0.24 kgm2【固Fig. 4.

Mass oflink I
Mass of link 2
Length of links l and 2
Vi町。出合icti。n of白e actuator
Moment of in，凶a of血e配同胞r

m2
L
Di
JM

視

M
AU

1
where 8 E :m,n+m- ,M(B）ε :m,(n+m-l)x（叶m-lJ, b(B, 9）ε:m,n+m 1,uε:m,n+m-1, and Tε:m,n ー 1.
Here, m-1 auxiliary com戸nents, z1, · · ·, Z m 1 , are必ded to thepa白 coordina蜘F・

p= (x1，·・·,Xn-1, s, Zt, · · ·, Zm-iJ=[:r?,i f
l

0.5
(32)

1

where：，；ε:m, - 飢dyεnlm. All the compon卸値of p including theauxiliary components should be independent, and白eJacobian m縮ix
1
Jε :m,(n+m- ）×｛叶m-lJ should be nonsingular. The equation of
motion is p紅白ioned 錨n

Ma Ha z+Ma Hp j-M a Hj8 +b a = T

Mp H0 i+Mp Hp j-Mp Hjf:i+bp = 0.

(33a)

一1

(-Mp Ha i+Mp Hjl:i-bp ).

T = {M a -M a Hp ( Mp Hp)

Mp }(Ha i- ajl:i)1+b a -Ma Hp (Mp Hp )- bp・

Step response of白e active ∞館脚neut (a) Active component：型. (b)Active component: x.

百e coordinate transformation加m joint space to operational space
is回P隠sented as
x = Lcos81 + Lcos(81+82)
官＝ Lsin81 + L sin(81 +82).
百1e Jacobian matrix is(34)

百E control law is a combination of PID feedback in (22) and thetorque computation-1

Fig. 6.

(33b)

If matrix Mp Hp ε nlmx m is nonsingi』lar,
量＝ ( Mp Hp ）

(b)

（35)

It has alm岡山e same form as (23).
V. EXPERIMENTS

A. Two-Degree-of-Freedom Man脚』ator
We con<h刷ed experiments using these con位。l me出ods with a
2・DOF horizontally articulat凶manipulator. Fig. 4 shows the ma­nipulator. The first axis ( 111) is a active joint and由e second axis
( 112 ) is a p邸sive joint.羽田active joint is driven by a de servo
motor with a harmonic-drive gear. The brake of the passive joint is
elec柱。magnetic. Fig. 5 shows the model of the manipulator. Table I
shows the par剖neters of 血E model. M and b in (I) are defined in(36), found at the bottom of the following page.
B. Control in Cartesian Space
The control scheme of Section II is demons廿·ated first. Cartesian
space is used as operational space.百1e origin is at the first joint.

J= LI1- sin81-sin(81 +82 ）I

81 +cos(81+82)

(37)

I

…sin(/:li +82 )cos(f'1+82) I /ト（38)

百町e are two ca回s: the case in which y is血e active component andx is the passive component, and 血e case in which x is曲e active
component釦d智is the passive component. From the state where
the manipulator is at問，st, a step change of the I官f釘ence is given
for the active component in each case. Fig. 6 shows the response.
百1e initial position is x = 0.4(m), y = O(m). In Fig. 6(a), y is
the active component and曲e reference is y = 0.05(m). In Fig.
6(b) x is the active component and the reference is x = 0.45(m).
百m f鈍dback gains are鎗t so 白紙血e pole of the system is a triple
root. The sampling interval is 2 ms with a 16-MHz i80386+8038 7
CPU. (In白e case of multiple degrees of企-eedom, (11) r，明日ires
2
2
3
2
2
(n -2n r+nr +7n -3nr+2r ] multiplications and the inversion
of a作－ r) x ( n -r) matrix. It would be desirable to develop
an efficient computation algori出m.) In the experimental 肥suit, themeasured value of the active component (solid line) converges tothe reference (dotted line）.百1e eπor from the reference aft紅白E
convergence is 0.14 mm in Fig. 6(a) and 0.08 mm in Fig. 6(b). Next,
a desired trajectory is assigned for the active component. Fig. 7 shows
the result of the tr勾ectory tracking. The active component increases
with constant velocity from the stationary state 飢d decreases withconst回t velocity m出e desir叫tr勾ectory. An abrupt change invelocity 田cu目 at the beginning of the 廿勾配tory and at the moment
the direction changes.η1e measured value (solid line) follows theIEEE TRANSACTIONS ON ROBOBTICS AND AUTOMATION, VOL. 9, NO. I, FEBRUARY 1993
E J

l

←

』 I

./

91

..,,._

0 !Oi

〆

、、

0.001"'
0.0

,
05

，
1.0

F

I

、：

xo= 0.35(m)YO =0 (m)YO =0以：mtsl
Yd= 0.2(m/s)

一一
Time [SJ 1 5

(a)

E J
訴

／

I

O 司451

札35 r
0.0

-

司、

I

/

、、

0.5

U

Fig. 8.

，｜

Motion of the manipulator.

司(b)Time [s] 1.5

Fig. 7.τracking of a desi問d町·aj配tory. （剖Active component: 9. (b) Active
G 沼田，ponent: .rdesired tr句ectory (dotted line) except just after those moments. The
stick diagram (Fig. 8) represents the motion of the manipulator when
the y component tracks a desired trajectory with constant velocity.
The initial acceleratio日is done wi出 the passive joint fixed. In Fig.
8. the y component increases constantly.

(a)C. Dynamic Singu/ariηIn the dynamic singular point, discussed in Section II-D, it isdifficult to apply the proposed control method. In the case of a 2-DOF
manipulator, the condition of dynamic singularity is MP HP = 0.
Fig. 9 shows the dynamic singular points of the manipulator used in
the experiments. MpHP = 0 at血ese points. It is necessary to avoid
these points in the control. In Fig. 9(a), l! is the active component, andin Fig. 9(b), .r is the active component. The dynamic singular pointsof Fig.倒的are not coincident with those of Fig. 9(b). In other words,
where one component is not controllable‘the other component is.

(b)

Fig. 9.

D. Modeling Error
This m巴thod depends essentially on the dynamic model of themanipulator. In the experiments of this paper, each parameter of themanipulator was calculated or determined experimentally in advance.
However, a load at the tip of the manipulator causes a change in the
dynamic parameters.
The control of the active components in the proposed method has
basically the same form as the computed torque method. In otherwords, the control of the active components is no less robust thanthe computed to叫ue method of a conventional manipulator having
an actuator for each joint. A high-gain feedback can suppress theE汀or caused by the modeling eπor if the change of the model is not
so large.

I !m1I 1 + ：！

M= I

3

唱

3

川

2

之

2

五

I

幽

苛

正

～

山内

/ -m2L �inB2ri,ti三 ：：＿ �m2L

b= I

On the other hand，出e passive components absorb the modeling
eπor. If the motion of the manipulator is simulated in advance, thetr句ectory of passive components deviates from the simulation. In
path-tracking control. the velocity profile along the path changes. The
proposed method is sensitive to the modeling eπor in this sense. The
authors expect that this method may be印me more effective if it isused together with a real-time parameter identification or an adaptive
control method.
We also investigated the robustness of the control method experi
mentally. A weight (O.,'i kg) is attached to the tip of the manipulator.
The same step-response experiments as in Fig. 6 are done using

2
L ＋ 川 L rnsB2 ＋ん,

ミ川 L ＋主川L｜

L囚 ation of dynamic singular points. (a) Active component: y. (b)Active component: .r.

lh！ m,L2 + !m,{ cosY, l
I
噌
I
�m2L'
2

in /i 2 Bi + D,li1

im2L九in 削 ｛

2

3

I

I

J

(36)

92

IEEE TRANSACTIONS ON ROBOBTICS AND AUffiMATION, VOL. 9, NO. I, FEBRUARY 1993

0.410
E
拠

0.400

0.390

。。

0.5

Ti皿e Isl

1.0

Ti皿e [s]

1.0

(a)

0.410
e
』

0.400

(a)

0.390
0.0

む5
(b)

Fig. l l. Suppression of tracking eπor. (a) Straight line path. (b) Circulararc pa山．

(b)Fig. JO. Path tracking motion. (a) Straight line pa出. (b) Circular arc pa曲 。
parameters without considering the weight. The e汀or of the active
components after the step response is x : 0,18 mm, y : 0.24 mm.
百1e increase in the error of the active component caused by themodeling error is small.
E. Path Tracking Control

The path－住acking con佐ol scheme of Section IV is applied to 血emanipulator. In this manipulator, the initial direction of the desired
path should be tangential to a circle centered at the first axis. First出e pぉsive joint is fixed and the active joint is accelerated. Path
tracking begins when the brake 時間leased. The desired pa
f Figs.
IO(a）叩d 1 l(a) is !liven as a straight line. In the case of(a), the pathcoordinate space is Cartesian space. The coordinate回nsformation is(37），組d the Jacobi組matrix (38). The desired pa血（a) is represented
as Xd
0.4(m). The desired pa
f Figs. IO(b) and l l(b) is given
as a circular 訂-c around the first axis. In the case of (b), the pathcoordinate space is a polar coordinate space whose origin is at由e
first joint.

曲。

=

曲。

r ＝

p= [r， φf ‘

山ヰ φ ＝ 81 ＋与(39)

羽e Jacobian matrix is

一
一

L

白日 唱A
rE・E・EEE’BEEL

J

=

-Lsin
1
2

些1
I
」

(40)

百1e desir官d p組1 is rd
0.4(m). Fig. 10 illus回.tes曲e results of由E
p柑トtracking control experime脳. The initial condition is Yo = 0 m,
動＝ 0.7m/s in加山cases.百e results（鈎，lid h凶） follow出e desired
paths (dotted lines). Trackings to the different p前hs are achieved from

Fig. 12Tracking of a composite path.

the same initial condition. The maximum deviation is 0.29 mm in Fig.
IO(a), and 0.72 mm in Fig. IO(b). Fig. II shows the response when
the initial position is not on the path. The deviations converge tozero by means of the feedback control. Fig. 12 shows tracking of apath that includes s位制ght and circular path segments. In this way, acomplicated path C銅 be composed of several simple path segments.
VI. CONCLUSIONS
A method to control a manipulator with passive joints in opera
tional space has been proposed. The equation of motion is represented
in terms of operational coordinates. The desired acceleration can begenerated at active components equal in number to the active joints
by using dynamic coupling among 血e coordin蹴S羽山method isextern命活to pa白・住acking con加，I. A path coonl1即ite system based on
白e desired path is defined. The equation ofmotion of the manipulator
is desαibed in旬nns of 山pa血 coordinates. The accelerationof the components normal to 血e desire4 pa白is controlled using
the dynamic coupling among components.τ'his in知m k関F陪血emanip凶ator on 白E desired pa白．百1e effectiveness of曲e me血od wasv出品ed by experiments usin? a乙DOF manipulator with a passive_
joint. One of 出e two coo吋泊ates in Car飽sian space is controlled
to follow a desired value. The exp釘iments also showed白at 白e

IEEE TRANSACTIONS ON ROBOBTICS A'iD At:TO\-IATION, VOL 玖'lO l 、 トEBRUARY 199393、

path of the manipulator can be controlled preじ1 討ely by u e of the
proposed method, Since this method uses cl川cd loop control、precise
tracking is possible in the presence of disturbances_ The path ofthe manipulator can be prescribed with this method and it may be
combined with obstacle avoidance or other path-planning algorithms
A〔K'iOWLED《j '.v\ENT
The authors would like to expre川 their appreciation to DL N
Oyama, Director of the Robotiじ%。cpartmenL Mechanical Engineer­ing Laboratory、for his constant suppo口They would also like tothank the members of the Biorobotics Division and the Cybernetics
Division, MEL, and K, Lynch, Carnegie Mellon Univer日ty, for theirassistance,
REFERENCES

，

，

，

、v

Inge Spangelo, J, Richard Sagli, and Olav Egeland
Abstract-In this work, we prove that if the manipulator Jacobian isproperly scaled. the largest singular value is bounded within a small in­terval close to unity. In this case, the inverse of the smallest singular value
is a good estimate of the condition number. This is useful in singularity
analysis. Bounds are derived for a general six-joint manipulator and for
the ABB IRb 2000 industrial robot in a c掛e study.
L [NTRODl'CTJOt,
The damped lea日－syuar白川lution ha, heen propo同d for mverse
kinematics with singularity robustness [ 1], [2], A critical point in
the implementation of this method is the selection of an appropriate
damping factor that gives a good compromise betw巴en accuracy and、feasibility o！ the ,olution
Nakamura and Hanaft】sa [ 11 proposed computing the damping
factor from the manipulability, Far from singularities. the damping
factor was set to Lero‘ while a nonzero damping factor was used
じlose to singularities where the manipulability approachesιroMaciejewski and Klein [3] proposed computing the damping factor
ぉa function of the smallest singular value of the Jacobian_ They
commented that the largest singular value of the Jacobian wasapproximately unity if the Jacohian was appropriately scaled, Thisobservation was also made by Wampler [2] who stated that the
Jarnbian should be s,aled "'ith the maxim山n reach of the arm. The
2 norm condition numtぅer of the Jacobian is the ratio between thelargest and the smallest singular alues_ If it can be proved that thelargest singular value is close to unity・it follows that th巴 inverse of
the smallest singular value i, a good estimate of the condition number.
The main contribution of this paper is the proof that the largest
singular alue is bounded close to unity when the Jacobian isproperly ,caled, We derive bounds for the largest singular value of the
日1anipulator Jacobian and discuss the effect of scaling the translations
It 1、父hown that if the translatiom are properly scaled, the largest
singular aluc is bounded within a small interval slightly larger thanunity. The analy川良i, done for a general six joint manipulator. andfor the ABB !Rh 2000 indu,trial rohoL

、

、－

、

II B 久仁 KGROLl1'D
The six dimensional vector of joint coordinates lぉdenoted q, The
three-dimensional vector of end c釘cctor velocity is denoted v, while
the three-dimensional angular velocity vector is denoted w_ The 6×6
Jacobian matrix J is defined h

[: l

、

= J（

仰

）
l
（

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1987

Bounds on the Largest Singular
Value of the Manipulator JacobianThi, matrix "as termed the partial velo�ity matrix by Wampler [2],If J is full rank 司 q can be found from (I) by Gau日ian elimination
In singular configurations, a solution will only exist if Iv Tw T) T E
Manuscript received November 25, 1991; revised June I, 1992. This work
was supported by the Royal Norwegian Council for Scientific岨d Industrial
Research through the Center for Robbotics Research at the Norwegian Instituteof Technology and by ABB Robotics,
The authors 町－e with the Division of Engineering Cybemetics, The Norwe­gian Institute of Technology, N-7034 Trondheim, Norway.
IEEE Log Number 9205082,

1042 296X/93$03.00 © 1993 IEEE