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528 IEEE TRANSACTIONSON ROBOTICSAN D AUTOMATION, VOL. 7, NO. 4, AUGUST 1991

Position Control of a Manipulator with PassiveJoints Using Dynamic Coupling

Hirohiko Arai, Member, IEEE, and Susumu Tachi, Member, IEEE

Abstrac t-This paper describes a method of controlling theposition of a m anipulator composed of active and passive joints.The active joints have actuators and position sensors. Thepassive joints have holding brakes instead of actuators. Whilethe brakes are released, the passive joints are indirectly con-trolled by the motion of the active joints using the couplingcharacteristic s of manipu lator dynamics. Wh ile the brakes areengaged, the passive joints are fixed and the active joints arecontrolled. The position of the manipulator is controlled bycombining these two control modes. This paper describes thebasic principle of the control method and the conditions thatensure the controllability of the passive joints. An algorithm forpoint-to-point control of the manipulator is also presented. Thefeasibility of the method is demonstrated by simulations for amanipulator with two degrees of freedom.

I. INTRODUCTION

HE number of degrees of freedom of a manipulator isT sually equal to the number of joint actuators. In orderto decrease the weight, cost, and energy consumption of amanipulator, various methods have been proposed for con-trolling a manipulator that has more degrees of freedom thanactuators. However, these methods require special mecha-nisms (e.g., drive chains, drive shafts, transmission mecha-nisms) in addition to the basic links and joints. In this paper,a method for controlling a manipulator having more jointsthan actuators without using additional mechanisms is pre-sented.

The dynamics of a manipulator has nonlinear and couplingCharacteristics. When each joint is controlled by a local linearfeedback loop, these factors result in disturbances. The elimi-nation of such disturbances has been one of the major prob-lems in the control of a manipulator [ 11- 5]. A design theoryfor a manipulator arm that has neither nonlinearity nordynamic coupling has also been proposed [6]. However, theforce of this disturbarice is available to drive a joint that initself does not have an actuator.

Vukobratovid and JuriZii [7] and Vukobratovid and Stokid[8] proposed an algorithmic control in which the states andgeneralized forces of the system are partially programmed,and the unknown states and unknown generalized forces are

Manuscript received August1, 1989; revised December18, 1990. Aportion of this paper was presented at the 20th International Sy mposiu m onIndustrial Robots, Tokyo, Japan, October 4-6, 1989.

H. Arai is with the Robotics Department, Mechanical Engineering Labo-ratory, AIST, MITI, 1-2 Namiki, Tsukuba, Ibaraki305, Japan.

S . Tachi was with the Robotics Department, Mechanical Engineering

Laboratory, AIST, MITI, 1-2 Namiki, Tsukuba, Ibaraki305,

Japan. He isnow with Research Centerfor Advanced Science and Technology, TheUniversity of Tokyo, 4-6-1 Komaba, Meguro-!a, Tokyo153, Japan.

IEEE Log Number 9101345.

determined by the conditions of dynamic equilibrium. Thismethod is applied to the synthesis of biped gait [7] and bipedpostural stabilization [8]. In the biped system, the degrees offreedom between the foot and the ground have no actuators,and they are controlled indirectly using dynamic couplingwith other powered degrees of freedom. Since dynamic equi-librium is represented as a second-order differential equation,a boundary condition is required to determine the state of thesystem completely. Vukobratovid et a l. used the repeatabilitycondition of the biped gait as the boundary condition.

This paper describes a method of controlling the positionof a manipulator composed of active and passive joints. Theactive joints have actuators and position sensors. The passivejoints have holding brakes instead of actuators. While thebrakes are released, the passive joints are indirectly con-

trolled by the motion of the active joints using the couplingcharacteristics of manipulator dynamics. While the brakes areengaged, the passive joints are fixed and the active joints arecontrolled. As the passive joints can be fixed by brakes, theboundary condition concerned with the active joints can bedetermined arbitrarily. The total position of the manipulatoris controlled by combining these two control modes. Thecondition that ensures controllability of the passive jointswhile released is obtained. An algorithm for point-to-pointcontrol of the manipulator is also presented. The feasibility ofthe method is demonstrated by simulations for a manipulatorwith two degrees of freedom.

11. PRINCIPLE F CONTROL ETHOD

Consider a manipulator with n degrees of freedom. We

assume that r ( r2

n / 2 ) degrees of freedom of the manipula-tor are active joints with actuators and displacement sensors,which is a typical structure of manipulator joints. The ( n - r )degrees of freedom are passive joints that have holdingbrakes instead of actuators.

When the holding brakes are engaged, the active joints canbe controlled without affecting the state of the passive joints.When the holding brakes are released, the passive joints canmove freely and are controlled indirectly by the couplingforces generated by the motion of the active joints. Theposition of the manipulator is controlled by combining thesetwo control modes.

The equations of motion of a manipulator can be written asfollows

M ( q ) 4 + 4 ) = U (1)where

b(q9 4 ) = h(q9 4 ) + r4 + g ( q )1042-296X/91/0800-0528$01.00 0 991 IEEE

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ARAl AND TACHI: POSITION CONTROL OF MANIPULATOR WITH PASSIVE JOINTS 529

and

are joint displacements,are generalized forces,are gravitational forces,are Coriolis and centrifugal forces,is the inertia matrix, andis the viscosity friction matrix.

The displacements of r joints, including all the ( n - r )passive joints, are selected from the elements of q and set asa vector $ E R ' (since r 2 n / 2 , r 2 n - r ) . When r > n- r , $ is composed of the displacements of the ( n - r )passive joints $pas E R"-' and those of the (2 r - n) activejoints $ E R Z r - n . When r = n - r , all the joints repre-sented by $ are passive. In addition, all the remaining( n - r ) oints are active, and their displacements are repre-sented as +ER"- ' . The generalized forces of the r activejoints are expressed as 7 ER' . When the passive joints arefree, the generalized forces of the passive joints are equal tozero. The elements of q and U are then rearranged as

Accordingly, M ( q ) and b ( q , q ) are also rearranged andpartitioned as follows:

n - r r

(3 )

When (2) and (3) are substituted for ( l ) , we obtain

Ml14+

M12$

+b , = 7

( 4 4

MZ14 MZ2$ + b, = 0 . (4b)The elements of M are functions of q . The elements of b

are functions of q and q . M , , , M 1 2 , M Z l r M Z 2 , , , and b ,can be calculated from the dynamic model of the manipulatorif the measured value of joint displacement and velocity ateach joint is substituted in q and q of (3). Furthermore,when desired values $d are assigned to the accelerations $,(4b) is considered a linear equation with regard to $. The

If we apply these generalized forces 7 to the active joints, theresulting accelerations will be +and $d .

In other words, while the passive joints are free, theaccelerations $ of r joints, including the ( n - r ) passivejoints can be arbitrarily determined by applying thegeneralized forces 7 to the r active joints. The accelerationsC of the remaining ( n - r ) active joints are determined by$d and cannot be adjusted arbitrarily. However, in addition

to the accelerations, the initial displacements and velocitiesare required to prescribe the motion completely. The initialdisplacements of the passive joints are the displacements atthe moment the brakes are released. The initial velocities ofthe passive joints are zero. The initial displacements andvelocities of the active joints are determined arbitrarily bycontrolling the active joints with the passive joints fixed.

While the passive joints are fixed, their velocities andaccelerations are zero. The displacements, velocities, andaccelerations of the active joints can then be controlledwithout affecting the passive joints.

By organizing the motion pattern using a combination ofthese two modes, the total position of the manipulator iscontrolled. In Section IV, n example of the organization ofthis pattern is described.

111. CONTROLLABILITYND OUTPUT ONTROLLABILITY

In this section, we present the theoretical basis of theproposed method from the standpoint of linear system theory[9]. First, we investigate the controllability of a manipulatorsystem with passive joints. The system is uncontrollable (inthe sense of linear system theory) if no gravity and nofriction act on the passive joints. In such a case, simultaneouspositioning of all the joints is impossible. Therefore, theholding brakes of the passive joints are necessary to ensurethe positioning capability of the manipulator.

Second, we investigate the output controllability of thesystem. The output is composed of displacements and veloci-ties of some of the joints. For output controllability, thenumber of joints we assign displacements and velocities

(output joints) must be less than or equal to the number ofactive joints. Including the released passive joints, we cansimultaneously control as many joints as the number of activejoints. Therefore, the number of passive joints should be lessthan or equal to the number of active joints.

Finally, we show the relationship between the linear ap-proximation and analytical expression by proving that theconditions of output controllability are equivalent to theconditions necessary to solve the generalized forces of theactive joints (i.e., M2 , is nonsingular).

coefficient matrix MZ 1 orresponds to the dynamic couplingbetween the accelerations of + and the generalized forces ofthe passive joints, and it depends the anddistribution of the manipulator. If M 2 1 s nonsingular, (4b)can be solved uniquely for + as

A . Linear ApproximationModel

At first, the equations of motion of the manipulator arelinearized to obtain a state variable representation. Since theinertia matrix M ( q ) is generally nonsingular, (1) is trans-formed as

( 5 ) q - M ( q ) - ' u + M ( q ) - ' b ( q , ) = 0 . (7)In order to ensure equilibrium while the passive joints are

7 = ( M , , - MllM;11M,2)$d b , - M , , M ; ' b , . ( 6 ) free, it is assumed that gravitational forces do not act on the

& = M - 11M22$d -

When ( 5 ) is substituted in (4a), we obtain

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530 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. I, NO . 4, AUGUST 1991

passive joints and that the friction of the passive joints can be ignored. When (7) is linearized in the neighborhood of theequilibrium point, the system can be represented by the following state equation:

X = AX + B r

I

where Next, we make the variable transformation x = Tz using Tn - r

2 = xz + Br (12)r n - r ( 9 ) where

ab1 ab1 ab1 ab1 ab1 ab1 ab1

a i 811, a4 a$ a i a$ a4--Nil - -NZ1 - - N i l - -NZ1 - N12 - -N22 - -N12 -

I, 0 0 0

0 0 0 0I0 0 I n - r 0

1M2184 + M228$

1

T-I = 17' 7' :2j.(11) = 1 0

-A = T - ~ A T

The state variables 64, 6$ , 6 4 , S$ consist of the deviationsof the joint displacements and velocities from the equilibriumpoint. The input is the generalized forces 7 of the activejoints (in this case, the number of inputs r and choice of 4,11, is arbitrary, so it is not necessary for $ to include all thepassive joints).

B . Controllability

Ml16i + M,26$MllW + M12WM2164 + M226$ .

2 = T- I x =

Here, the 2 n x 2 n matrix T is introduced as

0 N I 1 0 N I 20 N21 0 N 2 2 When the controllability matrix is calculated to investi-

T = (10) gate the controllability of system (12)

-y = [ B AB .. . A n - ' B]

ab1 ab1a i 811,

I r

M21 M22 0 0

and T is nonsingular. T- I is given byI , --Nil - -NZ1 * * * e

0 0 Mll M12

00 .. .. .

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r n ND TACHI: POSITION CONTROL OF MANIPULATOR WITH PASSIVE JOINTS

Since rank [ v ] 2 r < 2 n , system (12) is uncontrollable.Hence, system (8), which is equivalent to (12) , is alsouncontrollable. In other words, when the displacements andvelocities of n joints are simultaneously controlled with thegeneralized forces of r ( < n ) active joints, there exists adomain that the manipulator cannot reach by an input even inthe neighborhood of the equilibrium point.

C. Output Con rollability

Here, the output of the system (12)is taken as

y = cz

Y = [C =

r r n - r n - r ( 1 4 )

The displacements and velocities of r joints are the output. Ifthe system is output controllable., the existence of an input 7that transfers the output S$, S$ from an arbitrary value tozero is guaranteed. The complete condition for output con-trollability of systeGs (12)and (14)is that the rank of matrixN = [ C B C A B C Z CA"-' E1 equals 2 r .

( 1 5 )

From ( 15 ) , the condition of output controllability is equiva-lent to the nonsingularity of NZ1. ence, if NZ1 s nonsingu-lar, it is possible to set $ and $ at desired values and toperform local positioning in the neighborhood of the equilib-rium point.

Recall that r is the number of active joints. If r1 -

r ,we can select the r joints represented by $ so as to includethe ( n - r ) passive joints. In such a case, all the passivejoints can be positioned with the generalized forces of theactive joints. On the other hand, the displacements andvelocities of joints greater than r are not output controllablekespec tive of the output matrix since the rank of matrixN in ( 15 ) is less than or equal to 2 r . Hence, at least( n - 2 r ) passive joints cannot be controlled if r < n - r .

Furthermore, it can be proved that this condition agreeswith the condition that MZ1 must be nonsingular, and there-fore (4b) has a solution. This means that if and only if theaccelerations of r joints, including ( n - r ) passive joints,can be arbitrarily determined by the generalized forces of ractive joints without linear approximation, systems (12)and(14)become output controllable.

Proposition: When the n dimensional square matrix

n - r I

53

is nonsingular and the inverse matrix of M is represented as

n n - r

the complete condition that MZ1 s nonsingular is that NZ 1 snonsingular.

Proof: First, the sufficient condition is verified. Since NZ 1is nonsingular, when

n - r r

is postmultipled using both sides of (b)

Since both Q and M are nonsingular, QM - is also nonsin-gular. Accordingly, the vectors from the first to the ( n - r)throw of QM-' are linearly independent and N 1 2-N ll N ; l 1 N Z 2= N * ) is nonsingular. Further, if

n - r r

is postmultiplied using both sides of (d)

r n - r

Next, we postmultiply (f) by M to obtain

PQ = PQM-'M

NZlMll NZlM l,

Since

from (c) and (e), MZ1 N ; I . Accordingly, M ,, is nonsin-gular. With respect to the necessary condition, the exactsame proof can be made when M and N are exchanged.

IV . CONTROL LGORITHMIn controlling the position of a manipulator, several control

algorithms can be devised according to the way in which thetwo control modes are combined. As an example, point-to-point (PTP) control is considered here.

Since the passive joints are controlled with the brakesreleased (brakes-om) and the active joints are controlled withthe brakes engaged (brakes-ON), the control mode should be

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532 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 7 , NO. 4, AUGUST 1991

(sTART)

Fig. 1. Control algorithm in brake-OFF period

changed at least once so all the joints of the manipulator can

reach the desired position. It is considered difficult to controlboth the passive and active joints simultaneously to reach thedesired position precisely while the passive joints are free. Inthe simplest control algorithm sequence, therefore, first thepassive joints are positioned while the passive joints are free,then the remaining active joints are positioned while thepassive joints are fixed.

PTP control is possible with this sequence alone, but tomake it even easier, a brakes-ow period is added before thebrakes-OFF period. This brakes-oN period provides kineticenergy to the link mechanism through the active joints so thatthe passive joints can move easily and thus unnecessarymotion of the active joints is reduced. Consequently, theperiod of positioning is divided into the following threephases:

Phase I: $ joints are fixed (T o 5 < T , , brakes ON)Phase 11: $ joints are free (T , 5 t < T2, brakes OFF)Phase 111: $ joints are fixed (T 2 I < T 3 ,brakes ON)

In phase 11, r joints, including all the passive joints, arecontrolled along a desired trajectory. In phases I and 111, theremaining ( n - r ) active joints are controlled from the initialposition to the final position.

First, the trajectory and the generalized forces in phase I1are calculated off-line as follows (Fig. 1).

Step 1 Desired displacements? velocities and accelera-

joints, including all the passive joints, are pre-scribed. (In addition, $ J T, ) = gd(T2) 0 is aboundary condition because at the moment ofbrake switching, the passive joints must be atrest.)The initial displacements 4 ( T, ) and velocities

tions $ d ( t ) , $ J t ) , $ J f ) (TI I < T 2 )of r

Step 2:

4 ( T l ) of the remaining ( n - r ) joints are as-signed.Equations (5 ) and (6) are used to determine

4(T1 AT) and &(Tl + A T ) are determinedby numerical integration of 4 ( t ) . Here, AT isthe sampling interval.Steps 3 and 4 are repeated, and the generalized

forces 7 , ccelerations q , velocities q , and dis-placements q are determined for the rest ofphase 11.

The results of these off-line calculations are used for feedfor-ward compensation. Moreover, the boundary values of 4 and4 between the phases I, 111, and 11 are determined using thisresult.

When the control is executed in real time, the followingfeedback control is applied:

Step 3:

Step 4:7(Tl) and M,).

Step 5:

4; = 4, + K u ( l i d - 4) K J $ d - 4)

+ KiJ ( + d - / > d t ,

(K , , K , , K i : diagonal gain matrix) . (16)Here, $ d , lid,and 4, are the desired values of displace-ments, velocities, and accelerations given in step 1 and +and $ are the measured values of displacements and veloci-ties. Accelerations I obtained in (16) are substituted inof (6) and the generalized forces 7 are determined. In thiscase, the actual values for 4 and 4 can be obtained bymeasurement, and numerical integration is not required. From(44, (3 , 6), and (16)

( 4 d - 4 ) + q l i d - li) KP(& - $)

+ Ki/ (q d- 4) t = 0 . (17)

Therefore, - $ is guraranteed to converge to zero if K , ,K , , and K i are correctly chosen.

In phases I and 111, the control of the joint angles 4 issimilar to an ordinary manipulator. In phase I, 4 is con-trolled from the initial position 4(To) o 4(Tl) and & iscontrolled from 4 ( To )= 0 to &(Tl). In phase 111, ? iscontrolled from ? ( T 2 )to the final position 4 ( T3)and 4 iscontrolled from 4 ( T 2 ) o &(T,) = 0. Using the algorithmdiscussed above, the position of the manipulator can becontrolled between any two points.

V . SIMULATION

In order to confirm the feasibility of the proposed method,the control of a two-degrees-of-freedom manipulator (Fig. 2 )is considered. The mass distributions of links 1 and 2 areassumed to be uniform. The required actuator torque, thepositioning time, etc. are then evaluated. Specifically, (5 ) and(6) are expressed as follows:

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ARAI AND TACHI: POSITION CONTROL OF MANIPULATOR WITH PASSIVE JOINTS

lo 3

Eza0 102-

10'-

lo o

12

i n 1 2 ( B r a k e )

L =

I, 0.1 5

J o i n t 1 (Actuator 1

Fig. 2. Two-degrees-of-freedom manipulator.

0

e

e

ee 0 0

e 0

0 O

0

0 BrakeActuator

Brake ON

Brake OFF : 0

( + , + I : (0 ,01- ( 7 / 2 , 7 / 3 )

Fig. 3. Simulated motion of the manipulator.

TABLE ISIMULATIONESULTS

Positioning Time T3 - To I 1.78 sActuator Torque

Actuator VelocityActuator AccelerationBrake Torque

I?+I 5 2 . 3 N m

1@

1I

2.9 rad/s16 I 12.6 rad/s21T$ I 0 . 9 6 N m

The trajectory of $ in phase I1 is assigned arbitrarily. Theacceleration/deceleration curve is chosen to follow a sinu-soidal function in order to reduce the required actuatortorque. The maximum value of I$ I is assigned so that the

actuator torque is limited to a certain value, T2-Tl is ob-tained from the values of $ ( T I ) d $(T2) .The initial values of 4 and 4 (4(Tl), &Tl)) can be

arbitrarily selected. The trajectory of 4 is deteTined bysteps 2-5 in Section IV, and thereafter 4 ( T 2 ) nd 4 ( T 2 ) recalculated. The acceleration/deceleration of 4 along phases Iand 111 also follows a sinusoidal function.

An example of PTP control is illustrated using a stickdiagram of the two-link manipulator (Fig. 3). Since (18) and(19) have solutions only when M21 # 0, control of themanipulator is possible within the domain of - 2.30 < rC/