IEEE International Workshop o n Intelligent Robots and Systems
IROS '90

DYNAMIC CONTROL OF A MANIPULATOR WIT11 PASSIVE JOINTS- POSITION CONTROL EXPERIMENTS BY A PROTOTYPE MANIPULATOR Hirohiko Arai and Susumu Tachi
Robotics Department
Mechanical Engineering Laboratory, MITI
Namiki 1-2. Tsukuba, Ibaraki 305 Japan
Abstract: The authors propose a method of controlling
the position of a manipulator with passive joints
which have holding brakes instead of actuators. In
this method, the coupling characteristics of manipulator dynamics are used, and no additional mechanisms
are required. In this paper, the effectiveness of themethod is verified by experiments using a prototype
manipulator. The prototype is a two degree of freedom,
horizontally articulated manipulator. The first axis
is an active joint and the second axis is a passive
joint. While the brake of the passive joint isreleased, the passive joint is indirectly controlled
by the motion of the active joint, through the use ofdynamic coupling. While the brake is engaged, theactive joint is controlled. By combining these twocontrol modes. total position of the manipulator iscontrolled. The experiments show that the precise
positioning of the passive joint is feasible by useof the proposed method.

1 . Introduction
The number of degrees of freedom that a manipulator possesses is commonly equal to the number ofjoint actuators. In order to reduce weight. costand energy consumption of a manipulator, various
approaches have been proposed for controlling amanipulator which has more degrees of freedom thanactuators[ll . However, they require special mechanisms
in addition to the basic links and joints.
On the other hand, the dynamics of a manipulator
are non-linear and highly coupled. When each joint iscontrolled by a local linear feedback loop, thesefactors result in disturbance. The elimination of such
disturbances has been one of the major problems in thecontrol of a manipulatorC2-43. However, the force of
this disturbance is available t o drive a joint whichin itself does not have an actuator. These factors areutilized in several control methodsC51.
As a means of controlling a manipulator which has
more joints than actuators without additional
mechanisms, the authors have proposed a method ofcontrolling passive joints which have holding brakes
instead of actuators by using dynamic coupling. In aprevious paperC61, the principle of this method andthe condition that ensures controllability of thepassive joints was presented. An algorithm for point

to point control of the manipulator was alsopresented. The feasibility of the method was demonstrated by computer simulation of a manipulator
with two degrees of freedom.
In this paper, a 2-d.0.f.horizontally articulated
manipulator with a passive joint has been developed.
The efficiency of the proposed method was experimentally verified by this prototype manipulator
The control algorithm and the design of the control
system is also described.
2 . Prototype Manipulator

2.1 Hardware Structure of the Manipulator
The view of the prototype manipulator with twodegrees of freedom is shown in Fig.1. The first joint
is an active joint with an actuator, and the second
joint is a passive joint which has a holding brake
instead of an actuator. The lengths of both link 1 andlink 2 are 300mm. and the mass distribution of link 2is uniform. The actuator of the active joint is a 35W
DC motor with a 1/50 harmonic-drive gear. The motor
torque is controlled by motor current. A PWM current
amplifier is used. The maximum power of the amplifier
is + 7 5 V , + 4 A and 300W. The holding brake of thepassive joint is an electro-magnetic type with asingle friction disk. The static friction torque ofthe brake is 12". In order to achieve high-speed
operation, the brake is driven by an over-excitation
controller. An optical encoder is used to detect theangle of each joint. The encoder of the active joint
is ZOO,OOOP/R at the output shaft of the actuator. Theencoder of the passive joint is 24,00OP/R- 935 -Fig.1 The prototype manipulator2.2 Mathematical Model of the Manipulator
The prototype is modeled in Fig.2. Where,4(I: angle of the active joint: angle of the passive joint

ml(mz): mass of the link 1 (link 2)
L: length of the link 1lI(l2): distance between the 1st (2nd) joint andthe center of gravity of link 1 (link 2)
J1(Jz): moment of inertia around the center ofgravity of link 1 (link 2)
The equation of motion of the manipulator is written
as follows:

M l l~ t M 1 ~ ~ t D 1 ~ z 6 , ~ t D ~$+C,,=~ ~ C ' , r4' t c 4 (1)Mal 3tMzz$tDzri 6,'= 0(2)
=J
1tinl
1,'t
J~tm~l~'tm~L't2m~Ll~cos(ItJuMI
M I 2 = Mzl = Jntmnlz2tmzLlncos(/,Mn2 = J2tmz12'Dl = -2mzLlzsin(IDzll= mzL12sin(1,Dlzn= -mzLlzsin(IJ u : m n t of inertia of the actuatorc@ : viscous friction coefficient of the active joint

C4s : Coulomb friction torque of the active joint: actuator torque of the active joint
z4

Therefore, the control system of the manipulator
is composed of the following three levels.
1) Feedback control, which causes the angle andangular velocity of the passive joint to follow
desired values while the brake is released.
2) Trajectory planning and trajectory control ofthe passive joint while the brake is released.
3) PTP (Point-to-point) control of the manipulator
including switching of the brake and control ofthe active joint.
In this chapter, the basic principle of control of thepassive joint is introduced first, then the composition of control at each level is discussed.
3.1 Principle of Control
The coefficients of inertia (Mll,MIZ.MZI.Mzz). theCoriolis/centrifugal torques(Dll2d Ci).D12~&',DSII 6')
and the friction torques(c4 6,.CdS) can be determined
if the measured value of joint angle and angular velocity at each joint is substituted in 4 , 6,. (I and C',of the equations of motion (1) and (2). Furthermore.
when a desired value(=$d) is assigned to the acceleration $ . (2) is considered as a linear equation withregard to 4 . If )Isl f O . (2) can be solved uniquely as.
(3)

Where. it is assumed that each 1,ink is a rigid body,the friction torque of passive joints is negligible,
and the actuator torque can be controlled in proportion to the input voltage of the amplifier.

When (3) is substituted in (11,t D I I ~ ~ ,t &DIZZ&' t ~4 +dC $ s(4)If the torque r4 thus derived actuates the active
joint, angular accelerations 4 , i i , d arise. In otherwords, while the passive joint is free, the angularacceleration of the passive joint can be arbitrarily
determined by the torque of the active joint.
3.2 Feedback Control
In order to cause the angle and angular velocityof the passive joint to follow desired values while
the brake is released, T~ of equation (4) is taken as

Joint I ( Actuator 1

a feedforward term, and a feedback term is also added.
Then the following PID control law is applied.

Fig.2 Model of the manipulator
3 . Control System

T i =When the holding brake of the passive joint isengaged, the active joint can be controlled without
affecting the state of the passive joint. When theholding brake is released, the passive joint canrotate freely and is controlled indirectly by thecoupling torque generated by the motion of the active
joint. The position of the manipulator is controlled
by combining these two control modes.

T4+tKvG(C',d-Cil)tKp(I(%d-b'/)+Ki (I .f ( ( I d - (I)dt(5)

9 d , 3, d and $,, are desired values of the angle,angular velocity and angular acceleration of thepassive joint. (I and C', are measured values of theangle and angular velocity. When the torque calculated
by equation (5) is applied at the active joint, thefollowing relation can be obtained from (1). (2). (4)- 936 -

and (5)(6)$d-$is guaranteed to converge to zero if K
P G ' Kv(l,and Ki are selected appropriately.
Feedback of the angular velocity of the passive
joint is included in control law ( 5 ) . However, sincethe resolution of the encoder to measure Cr, is lower
than that of 9 , a quantumizing error occurs when Ci,
is determined from the numerical differentiation of 9 .
Moreover, because of the compliant factor at the gear' between the encoder and the motor. oscillation arises
when the feedback gain is high. In order to increase
the stiffness while retaining stability, a new control
law was tested in which proportional/ integral feed,back of c/, is combined with velocity feedback of 9 .
If the centrifugal torque is negligible and thevalue of M Z I scarcely changes, integrating both sidesof (2) produces:(7)

When d d is the desired angular velocity of the active
joint, the new feedback law is in the form of:I t may produce a result approximately equivalent to
the feedback of equation (5). Note that d d is not an
arbitrary desired value; it is a value obtained byintegration of $ d in equation (3).
The method mentioned above can be interpreted asa special case of the method in which the value of Cjr,
is estimated by a state observer. State equations ofthe system are obtained from linear approximations of(1) and (2) as,i=Axtbt(9)

Of the state variables, y=C 4 , Cr,, $IT are assumed to
be detected as outputs, and a minimal order observerwhich estimates $ is designed. ( 4 is obtained fromthe numerical differentiation of 4 . )t = 2z t 6y t T T$=ztly(10)l=Cll,1 2 . la] are assigned for design parameters Eachcoefficient is determined by Gopinath's method asfollows.
iz-12, 6=[-1112, -1z2, -11 t (Nz 1 - N I 113) c 4 -1 z 1 3 1?=NZI-13NI1(11)Since the pole of the observer is -l2. it is stable if12)O. When 11=0 and ~ ~ = N ~ I / N I ~ ( = - M respectively,~~/M~~)

k = -1zz -1z2(I, + -12MPMz z

1$3(12)

So 4 can be estimated from c/, and d only.
The observer (12) can be represented as a transfer
function in the following form.

Consequently, 5 is obtained as sum of the differentiated value of Cr, filtered by the first order LPF
with a cut-off frequency of l2/2r, and - ( M z I / M Z Z ) $3
filtered by the first order HPF of the same cut-off
frequency. If lZ=O. G=-(M21/M2z)d . So the same value
as with equation (8) is fed-back. However, when 1 ~ = 0 ,the deviation of 5 from 6 does not converge to zeroand error arising from the centrifugal torque andtemporal change of M z l are accumulated. In the control
law ( 8 ) . d d is obtained by integrating (3). in whichthese factors are taken in account, so that errors
from these factors may be compensated.
3.3 Trajectory Control of the Passive Joint
Positioning of the passive joint is achieved along
a desired trajectory while the brake of the passive
joint is released. In this section, design of desired
trajectory and trajectory control for the passive
joint are described. The initial value of the joint
angle is taken as G o and the desired value,,as ",end
The desired trajectory q~~ (TIS t S T 2 ) should satisfy
the following three conditions:a) UJ~(TI) = G O . Gd(Tz) = $'endb) & , ( T i ) = cl;d(Tz) = 0C) $d(t)(TIStSTz)
is a finite value.
Of these, a) is the condition for executing positioning along the trajectory: b) is a boundary condition
because at the moment of the brake switching, the passive joints must be at rest; and c) is the condition
that the desired trajectory can be realized physically.
Furthermore, the lag in brake operation time (1020ms for ON-OFF, 20-50ms for OFF-ON) is considered
and the following condition is added:d) &d(t) = 0 (TIStST;, T;StlTz)
The brake is released at TI and the brake is engaged- 937 -at Ti. T;-T1=POmsand Tz-T;=GOms.
The following time function is used as thetrajectory which satisfies all the above conditions.

$dand t,,(t).
@ The values of $,, and d d after the sampling interval AT are determined by numerical integration of$d
and $ d .
@ Steps @ and @ are repeated and @d(t)and d d ( t )are determined within the period of TISt STz

3.4 PTP Control
In addition, a maximum value t,+max is defined for thethe active joint torque. Tz-T, is determined so thatthe inertial part of the active joint torque does notexceed this value. From equation (4).
U 11Gdmax = (M1Ail1 - Y r l M ~ 1 ) 7 ~ m a x
Then,(15)

T;-T; = 12n ((I,end-J/o)/GdmaxI"2(16)In order to perform positioning between two arbitrary points, two control modes (brake ON and brakeOFF) are combined and a motion pattern is compiled.
Since the passive joint is controlled with the brakereleased and the active joint is controlled with thebrake engaged, the control mode should be changed atleast once before both joints of the manipulator are
set at a desired position. Here. mode switching isperformed twice and the period of positioning isdivided into the following three periods:

An example of a designed trajectory is shown in Fig.3.
In order to cause the passive joint to follow thedesired trajectory, feedforward calculation ofequation (4) and feedback calculation of equation ( 8 )are executed based on the desired values ( d J d , ( i d ,and the measured values ( 4 , $ , (I,, $1. and t,+ isdetermined. If the values of 4 d(t) and of 7 $ (t)(TIStSTz)are calculated in advance by the following
procedure, real time control requires only calculationof (8) and the sampling time can be reduced.
@ The desired trajectory of the passive joint (I,d(t)(TlltSTz) is prescribed, then fiJd(t) and $d(t)are determined.
0 The initial state of the active joint ,+(Ti). $(Ti)is assigned.
0 Equations (3) and ( 4 ) are calculated to determine

J/.nd(I,O0I Gdmaxl
0-I
TI'TzTimeFig.3 Desired trajectory of the passive joint
Ti

Ti 'ActiveJoint

CurrentPosition(90, cbo)

Ac tuato]
SensorDesiredPosition
Joint(,+end*&ild)

Fig.4 Control system of the manipulator- 938 -

( i )( ii)

T n 5 t 5 T l , brake ON (fixed)

TIItSTI,brake OFF(re1eased)( i i i ) T z l t l T a . brake ON

In period ( i ) , initial acceleration of the active
joint is performed; in period ( i i 1 , positioning of thepassive joint is performed; and in ( i i i ) . positioningof the active joint is performed.
The initial angles of the active and the passive
joints are ( $ o , G o ) . and the desired angles are(d&, &n,j). Positioning of the passive joint from G oto (//end in period ( i i ) is discussed in section 3.3. Inperiod ( i ) , control is performed in which the desired
state is d=dd(T1). d=dd(T1) from the state $ = d oarid do=(). In period ( i i i ) . control is performed inwhich the desired state is $=dend.d=O from thestate d=dn(T2),d=dd(Tz). For period ( i ) and ( i i i ) ,the maximum value of I $ I is determined based on themaximum torque of the active joint the same as withperiod ( i i ) , and the trajectory with a sinusoidal
acceleration/deceleration pattern is designed. Since
the initial state of the active joint $ d ( T 1 ) , dd(Tl)in period ( i i ) can be selected arbitrarily, it was
selected by trial and error so that the time T3-To for
positioning would be minimum. Feedback control ofactive joints in periods ( i 1 and (iii) is the same ascontrol of common manipulator joints.
The control system comprised of the three levels
mentioned above is illustrated in Fig.4.

4 . Experiments
The values of parameters in the equation of motion(1) and (2) of the prototype manipulator are shown in,D21 were determined by
Table 1. M I Z .Mz1, M Z Z .D1 z ,DIZZ
calculation from the following conditions: mz=l.l(kg).
L=0.3(m). 1~=0.15(m) and the mass distribution of link
2 is uniform. In addition, C
was determined fromthe step response,
the torque measurement, c$and M I I from the angular amplitude with respect to thesinusoidal input torque. The relation between actuatortorque and amplifier input voltage was also verifiedexperimentally.
The control system described in Chapter 3 wasimplemented on a personal computer (1GhiHz 80386 CPU tr",bTable 1 Parameters of the manipulator
M110.61 t 0.090 sin$ Ckgm']
MI20.030 t 0.045 coscl, [kgm']
Mz1
0.030 t 0.045 cosG [kgm']
Mzz
0.030
Ckgm'l
D112-0.090 sin(/,Ckgm'lDIZZ -0.045 sin(/,Ckgm21D Z I I 0.045 sin(/,Ckgm'lc($2.2
Ckgm2/slc@s 4.3CNml

ActiveJoir,tPersonal +Converter
Computer-1~ServoAmp. -+D/A--- @r Encoder

CLPassive

PowerPack-+~Pulse(8038GCPU +- CounteriI

Joiri:

t80387FPU)
Relay_Brake- Encoder- 939 -_

was 1.7X lO-'rad. Since the resolution of the encoder
was 2.6xlO-'rad. a repetitive precision close to theoptimum could be obtained.
5. Conclusion
In this paper, a horizontally articulated
manipulator with two degrees of freedom was developed,
which has a passive joint comprised of a holding brakeand an encoder at the second axis. Fundamental experiments on position control of the manipulator using
dynamic coupling were performed. It was confirmed that
positioning of the passive joint with high repetitive
precision was possible by the proposed control method.
This method depends essentially on a dynamic modelof the manipulator. In the experiments of this paper,
each parameter of the manipulator was calculated ordetermined experimentally in advance. However, thismethod may become more effective if it is used
together with a real-time parameter identification or
adaptive control methodC71.

C31 E.Freund. "Fast Nonlinear Control with Arbitrary
Pole-Placement for Industrial Robots and Manipulators," Int. J. Robotics Research, Vol. 1 . No.1,pp.65-78, 1982.
C41 M.Vukobratovi6 and D.StokiC , "One Engineering
Concept of Dynamic Control of Manipulators,"
Trans.ASME.J.DSMC. Vol.103.No.2,pp.108-118. 1981.
C51 M.VukobratoviC. and D.JuriciC. ,"Contributionto theSynthesys of biped gait." IEEE Trans. BME, Vo1.16.
No.1. pp.1-6. 1969.
161 H.Arai and S.Tachi, "Position Control of aManipulator with Passive Joints Using DynamicCoupling," Proc. 20th ISIR. pp.617-624, 1989.
C71 J.J.Craig et.al., "Adaptive ContrQl of Mechanical
Manipulators," Proc. IEEE Conf. Robotics andAutomation, pp.190-195. 1986.

References

C13 S.Hirose and Y.Umetani. "The Development of Soft
Gripper for the Versatile Robot Hand," Proc. 7thISIR. pp.353-360. 1977.
121 J.Y.S.Luhet.al. ."On Line Computational Scheme for
Mechanical Manipulators." Trans. ASNE, J. DSMC.
V01.102. No.2, pp.69-76. 1980.

00

0.1

0.2

0.3

0.4'

0.5
TimeCsl(aIExperimenta1 result1(ICradl

0.5
0 5,t'0103-.

0.1

0.2

0.3

0.4

000.1

0.2

0.3

0.4(b) Desired trajectory
Fig.? Trajectory tracking of the passive joint

0.5
TimeCsl(a)Experimental result by feedback(5)Brake ON : O
Brake 0 F F : Olt@I[rad 1' 0.1

0'
00.2

0.3

0.4

0.5

TimeCsl

0.5

TiaeCsl(b)Experimental result by feedback(8)Fig.6 Step response of the passive joint- 940 -