Robust Arm Configuration of Manipulator Mounted on Flexible Base

IEEE TRANSACTIONS ON ROBOTICS, VOL. 20, NO. 4, AUGUST 2004 781

Robust Arm Configuration of ManipulatorMounted on Flexible Base

Jun Ueda and Tsuneo Yoshikawa

Abstract—In this paper, the robustness of a manipulator mounted on aflexible base with a task-space feedback control to a fixed desired point isconsidered. We define the robust arm configuration (RAC) , which is a spe-cial configuration where the linearized system is positive real. Lyapunovindirect method and the passivity theory guarantee a local asymptotic sta-bility of the original nonlinear system. A finite but high closed-loop gaincan be applied in the neighborhood of the RAC without considering thebase flexibility, i.e., an additional sensor or a solution of whole inverse dy-namics is not necessary. Considering the positive semidefiniteness of theresidue matrices, a measure is proposed that measures the distance fromthe RAC. This measure represents the controllability of the manipulator it-self, and does not depend on the underlying control law. The validity of theproposed approach is confirmed by a numerical example and experiments.

Index Terms—Base flexibility, flexible robots, mechanical resonance,modal analysis, robust control.

I. INTRODUCTION

The demand of high speed and accurate assembling is increasingwith the advance of industrial products. A precise positioning of theend-effector to a fixed desired position is fundamental and important todeveloping a high-speed assembly system, where a manipulator picksup a part from a pallet, and places it to a workpiece. In this system,the positioning error of the pallet or the workpiece is unavoidable andshould be compensated. In order to achieve this task, a task-space feed-back control, using the positioning error between the end-effector andthe desired position, is more desirable than a joint variable controlbased on the solution of the inverse kinematics. In a joint variable con-trol, the error is unavoidable in the inverse kinematics, and the precisemeasurement of the global position of the parts and the workpiece isdifficult. In contrast, a precise measurement of the positioning error re-quired for a task-space control is feasible, using a narrow positioningsensor attached to the pallet or the end-effector.

One of the primary factors of limiting the bandwidth of this kindof system is the flexibility of the base on which a tool or a manipu-lator is mounted. When the joint sensors and actuators of the manip-ulator are located at the same place, the system looks collocated. Butit is not collocated, due to the flexible base. This problem of the flex-ible base is, therefore, hardly recognized. Although the stability of theend-point feedback control for a rigid manipulator is guaranteed [1], itsapplication for a flexible manipulator is not straightforward [2]. Muchresearch has been presented on flexible link and elastic joint robots,with respect to joint trajectory control [3], stability of joint-level con-trol [4], [5], and feedforward tracking control [6]. However, the stabilityof the end-point feedback control still remains a difficult problem. If atask-space control is applied to the flexible-base manipulator withoutpaying attention to the base flexibility, the closed-loop system can be

Manuscript received December 3, 2002; revised October 21, 2003. This paperwas recommended for publication by Associate Editor F.Wang and Editor A. DeLuca upon evaluation of the reviewers’ comments. This paper was presented inpart at the IEEE International Conference on Robotics and Automation, Wash-ington, DC, May 2002.

J. Ueda is with the Graduate School of Information Science, Nara Insti-tute of Science and Technology (NAIST), Nara 630-0192, Japan (e-mail:[email protected]).

T. Yoshikawa is with the Department of Mechanical Engineering,Kyoto University, Kyoto 606-8501, Japan (e-mail: [email protected]).

Digital Object Identifier 10.1109/TRO.2004.829482

easily destabilized. One possible solution is stabilizing by the inversedynamic model, but this approach needs a complex controller and pre-cise measurement of the deformation of the flexible part using addi-tional sensors, such as strain gauges or accelerometers.There is a long history of simultaneous optimization of structure and

control to solve the interaction problem between a flexible mechanismand a controller [7]–[10]. In simultaneous optimization, in spite of itsstraightforwardness, the ambiguity in mixing both parameters has beenpointed out [12]–[14], since this method heavily depends on numericalcalculation. In addition, the convergence of parameters is not guaran-teed, and few theoretical clarifications are given for its optimality. Re-cently, to overcome these problems, mechanical-design methods basedon passivity [11]–[14] have been studied. The basic idea of this ap-proach is separation of mechanism and control design. It has been in-dicated that making the region of the frequency domain where pas-sivity is satisfied as large as possible improves the robustness of the me-chanical system itself. The in-phase design [11], [15] for single-inputsingle-output (SISO) systems is essentially based on this passivity cri-terion; however, multiple-input multiple-output (MIMO) systems, suchas manipulators, are hardly examined.In this paper, the relation between the arm configuration of a manip-

ulator mounted on a flexible base and the robustness, i.e., a passivity-based evaluation of the arm configuration, is investigated. We focus ona task-space feedback control with Jacobian transpose to a fixed desiredposition, based on the positioning error between the end-effector andthe target. We show that there exist special configurations for whichthe linearized dynamics is positive real. We define this configurationas the robust arm configuration (RAC) [16], where high robustness canbe obtained. The inverse dynamics solution and measuring the flexibledeformation are not necessary, but a simple positioning error feedback,e.g., a controller under assumption of a rigid manipulator, is enough.A measure is proposed which indicates the distance from the existingconfiguration to the RAC. This measure represents the robustness ofthe manipulator itself, and does not depend on the underlying controllaw.

II. RAC OF MANIPULATOR

A. Flexibility of the Base

A typical situation we are going to study is given in Fig. 1. A flexibleassembly cell which consists of a 2-degrees-of-freedom (DOF) planarrigid manipulator and two pallets on moving tracks is considered. Asshown in Fig. 2, the parts are taken out from the parts feeder and areattached to the workpiece. The base driven by a ball screw on whichthe manipulator is mounted is considered as a passive viscoelastic joint.Since the control bandwidth of the ball screw is not high comparedwith the manipulator, the ball screw is positioned beforehand and isnot driven during the assembly. During a motion, the root of the basetwists, due to the flexibility of the ball screw and the guide rods thatsupport the base.In general, it is very difficult to attach additional sensors to mea-

sure the base flexibility. Compensation is through a task-space feed-back control. The positioning error between the pallet and the end-ef-fector is measured directly, using a narrow visual sensor attached to thepallet or the end-effector. If the servo gain of the task-space control isincreased, the full closed loop is easily destabilized.

B. Dynamic Equation

The system consists of an m-DOF flexible base and an n-DOFnonredundant manipulator, which includes the system of Fig. 4 as aspecial case.

1042-296X/04$20.00 © 2004 IEEE

782 IEEE TRANSACTIONS ON ROBOTICS, VOL. 20, NO. 4, AUGUST 2004

Fig. 1. Flexible assembly system.

Fig. 2. Flexibility of the base.

�� denotes joint torque, and qqq the joint axis displacement vector. Thedynamic equation is as follows:

�� =MMM(qqq)�qqq + hhh(qqq; _qqq) (1)

�� = [��Tp ; ��Ta ]

T 2 <N (N = m + n) where ��p =[�1 � � � �m]T is torque generated by passive viscoelastic joints,��a = [�m+1 � � � �m+n]

T is actuator torque generated by motors.Similarly, qqq = [qqqTp ; qqq

Ta ]

T 2 <N where qqqp = [q1 � � � qm]T andqqqa = [qm+1 � � � qm+n]

T . The subscript p denotes the passiveviscoelastic joints and a denotes the active joints. Note that qqqpcannot be measured directly. MMM(qqq) 2 <N�N denotes the inertiamatrix and hhh(qqq; _qqq) are the centrifugal and the Coriolis factor. Theeffect of the gravity is neglected, as it is considered unimportant. Atask-space control by measuring the positioning error is necessary tocompensate the uncertainty of the inverse kinematics of a manipulatorand uncertain position of the moving tracks. In this paper, we focuson a task-space feedback control of the end-effector using Jacobiantranspose. Let rrr = rrr(qqq) 2 <n, the position of the end-effector,rrrd as the desired position of rrr, and �qqqp = [��1 � � � ��m]T as theequilibrium points of qqqp. rrrd is realized by qqqd = [�qqqTp ; qqq

Tad]

T , where�qqqad = [�(m+1)d � � � �(m+n)d]

T , and it satisfies rrrd = rrr(qqqd). Letrrrv = rrrv(qqqa) be the position of the end-effector when the displacementof the base is zero, so that rrr(qqqd) = rrrv(qqqad). It is assumed that thetorque �1; � � � ; �m generated by passive joints are working to settle tothe equilibrium points ��1; � � � ; ��m

�i = kpi(��i � �i)� kvi _�i; i = 1; � � � ;m (2)

where kpi and kvi are the stiffness and the viscous coefficient, respec-tively.

Hereafter, we examine the local asymptotic stability of the nonlinearsystem (1) around rrrd (around �qqqp, qqqad for joint space) by its linearizedmodel. In this regard, the centrifugal and Coriolis factor can be ignored,since these terms are not related to the stability. Substituting (2) for (1),we obtain

0

��a= MMM�qqq +DDD _qqq +KKK(qqq � qqqd) (3)

where DDD = diag(kv1; � � � ; kvm; 0; � � � ; 0), KKK = diag(kp1; � � � ;kpm; 0; � � � ; 0) are the stiffness and the viscous matrix, respectively.Replacing xxx = [_qqqT ; (qqq � qqqd)

T ]T and yyy = rrr(qqq) � rrr(qqqd),n-inputs/n-outputs state-space representation in the work coordinatesystem is obtained

PPP (qqqd; s) = CCC(sIII �AAA)�1BBB (4)

where

AAA =�MMM�1DDD �MMM�1KKK

III OOO

BBB =MMM�1 OOO

IIIOOO

JJJTv

and CCC = [OOO JJJ ]. JJJ = @rrr=@qqqT 2 <n�(m+n) and JJJv = @rrrv=@qqqTa 2

<n�n are Jacobian matrices.We have assumed that this flexibility is not measured, then ��a is

calculated as

��a = JJJTv fff (5)

where fff denotes the control input force in the taskspace calculated by afeedback controllerKKKc(s) : fff = �KKKc(s)yyy. Note that we usePPP (qqqd; s)when we emphasize that PPP (s) is a function of qqqd, otherwise we usePPP (s).

C. Modal Analysis

By modal analysis, PPP (s) can be expressed as a linear sum of therigid mode and them vibration modes.AAA has in total 2(n+m) poles,2n of these poles are zero, corresponding to the rigid mode, and 2mconjugate complex poles correspond to the vibration modes.Let �0 = 0, �i, ��i; i = 1 � � � ; m be 2m + 1 distinct eigen-

values of AAA. �0, �i, and ��i correspond to the rigid mode andthe ith vibration modes, respectively. We define matrix UUU and VVV

using uuu which satisfies AuAuAu2j�1 = �0uuu2j�1 = ooo, j = 1; � � � ; n;AuAuAu2j = uuu2j�1; AuAuAu2(n+i)�1 = �iuuu2(n+i)�1, i = 1; � � � ; m;

AuAuAu2(n+i) = ��iuuu2(n+i) whereUUU�= [uuu1 � � � uuu2n uuu2n+1 � � � uuu2(m+n)]

and VVV ��= col[vvv�1; vvv

�

2; � � � ; vvv�

2(m+n)] = UUU�1. vvv� denotes the complexconjugate transpose of vvv. Applying UUU , VVV to AAA, the modal analyzedtransfer function [17] of PPP (s) is obtained as

PPP (s) =1

s2RRR0 +

m

i=1

1

s2 + 2�i!is+ !2iRRRi (6)

RRR0 =

n

j=1

CuCuCu2j�1vvv�

2jBBB = JJJvMMM�1JJJTv (7)

RRRi = � 2< ��i CuCuCu2(n+i)�1 vvv�2(n+i)�1BBB : (8)

RRR0 corresponds to the rigid body mode, and RRR0 is positive semidefi-nite. MMM = [Mkl] 2 <n�n(m+ 1 � k; l � m+ n) is a partial matrixof MMM . RRRi is called residue matrix. Note that RRR0 is positive semidef-inite and rank (RRRi) = 1, at most, from (8). In addition, !i = j�ij,�i = �<(�i)=j�ij is obtained, where �i and !i represent the dampingcoefficient and the natural frequency, respectively.


D. Positive Realness of Transfer Function Matrix

The robustness of PPP (s; qqqd) is evaluated based on its positive real-ness. LetGGG(s; qqqd) be the transfer function from the control input fff tothe velocity _yyy

GGG(s; qqqd)�= sPPP (s; qqqd): (9)

Similarly, we useGGG(qqqd; s) when we emphasize thatGGG(s) is a functionof qqqd, otherwise we use GGG(s).

The following theorem is obtained.Theorem: GGG(s) is positive real if and only if the following condi-

tions 1 and 2 are satisfied for 8i : i = 1; � � � ; m, 8j : j = 1; � � � ; n:

1) RRRi = RRRTi ;

2) �ijj > 0 where RRRi = [�ikl]; k; l = 1; � � � ; n.Proof: Since it is known that the sum of positive real functions

is positive real, we consider the positive realness of each individualmode. From (6), we define GGG0(s) and GGGi(s) which correspond to therigid and ith vibration mode

GGG0(s)�=

1

sRRR0 (10)

GGGi(s)�=

s

s2 + 2�i!is+ !2iRRRi: (11)

Since RRR0 = RRRT0 , we have GGG0(j!) +GGG

T0 (�j!) � OOO. Hence, GGG0(s)

is lossless and positive real.Next, the condition for the positive realness ofGGGi(s) is

GGGi(!)�=GGGi(j!) +GGG

Ti (�j!)

=2�i!i! RRRi +RRR

Ti + j! !2i � !2 RRRi �RRR

Ti

(!2i � !2)2+ 4�2i !

2

i !2

� 0 8! 2 <: (12)

When strict inequality holds, GGGi(s) is said to be strictly positive real.A necessary and sufficient condition for the positive semidefinitenessof a Hermite matrix GGGi(!) is that all principal minors are nonnegative[18].

Considering rank (RRRi) = 1, it can be said that all the determinantsof principal minors larger than 3 � 3 are zero, since the rank of allminor matrices is less than two. Consequently, it is enough to examinethe 2� 2 principal minor. Since the rank of any 2� 2 principal minor�i of RRRi is one, it can be described as

�i =a �a

�a ��a(13)

where a, �, and � are real numbers.Substituting (13) for (12), the necessary and sufficient condition of

positive semidefiniteness becomes

det2�i!i! �i +�T

i + j! !2i � !2 �i ��Ti

(!2i � !2)2+ 4�2i !

2

i !2

= det1

�(!)

g11 g12

g21 g22

= � 4�2i !2

i !4 + !

2!2

i � !2 2

(�� )2a2

� 0 8! 2 < (14)

where

�(!) = !2

i � !2 2

+ 4�2i !2

i !2 (15)

g11 =4�i!ia!2 (16)

g12 =2�i!i(�+ �)a!2 + j! !2

i � !2 (�� )a (17)

g21 =2�i!i(�+ �)a!2 � j! !2

i � !2 (�� )a (18)

g22 =4�i!i(��a)!2: (19)

It is clear that the equality of (14) holds if and only if � = �.GGGi(j!)can not be strictly positive real, since the left-hand side of (14) is non-positive, i.e., no qqqd satisfies GGG(s; qqqd) + GGG(s; qqqd)

� > 0, 8<[s] > 0.Consequently, the necessary and sufficient condition of the positive re-alness (passivity) ofGGG(s) is that all residue matricesRRRi are symmetricand positive semidefinite;RRRi = RRR

Ti (condition 1) andRRRi � 0.

Moreover, rank (RRRi + RRRTi ) = 1 holds if RRRi = RRR

Ti , then all 2 �

2 principal minor of GGGi(j!) + GGGTi (�j!) equal zero if RRRi = RRR

Ti .

Namely, the condition RRRi � 0 is equivalent to the condition that alldiagonal elements ofRRRi is nonnegative (condition 2).

Remark: Condition 2 of the theorem is equivalent to the conditionthat all diagonal elements ofPPP (s) should be in-phase. However, it wasshown that no qqqd gives strictly positive realGGG(s). This fact shows thatthere is no in-phase design [11] for MIMO systems, although it wasproposed and shown to be effective for SISO systems.

E. Definition of RAC

Definition: An arm configuration qqqd which satisfies

�min GGG(j!; qqqd) +GGGT (�j!; qqqd) = 0 (20)

is called RAC, where �min(GGG) denotes the minimum eigenvalue ofGGG.For this configuration, conditions 1 and 2 of the theorem are satisfied.

The RAC is a special configuration, for which the linearized system ispositive real (passive). Note that the dimension of the set of all RACsis not equal to that of the workspace. In the example of the flexibleassembly system shown later, the configurations draw a 1-DOF curvedline on the workspace.

F. Stability in the Neighborhood of RAC

It is well known that the global stability of task-space control is guar-anteed for rigid manipulators [1]. The results obtained in the subsec-tion D suggest that this global stability can not be expected for flex-ible-base-mountedmanipulators with task-space control, since even thelinearized mechanism is not positive real over a wide working range.However, it implies that a local asymptotic stability of the original non-linear system can be guaranteed with a finite but high feedback gainwithin a small area, including the RAC. The proof is straightforwardfrom Lyapunov indirect method and the passivity theory.A stable positioning controller with finite velocity gain can be

designed easily around the RAC, since the dynamics is smooth. Thepassivity theory guarantees the stability with an infinity velocity gainexactly on the RAC. The Lyapunov indirect method gives the localasymptotic stability of the original nonlinear system. The existenceof the attraction region is obvious, as long as a finite gain is used,although the bounds of this region depend on the feedback gain. Asa result, a high positioning performance can be obtained withoutconsidering the base flexibility, i.e., without any additional sensor orsolution of a whole inverse dynamics.

III. ROBUSTNESS MEASURE OF ARM CONFIGURATION

A. Robustness Measure Based on Residue Matrix

In this section, a measure that indicates a distance from the RAC isproposed. Recall that if the rank (RRRi+RRR

Ti ) � 2, there exist only min-

imum and maximum eigenvalues. The system is positive real if rank(RRRi +RRR

Ti ) = 1 and �min(RRRi +RRR

Ti ) � 0 (in fact, this value is neg-

ative semidefinite, as will be shown below). We propose the followingrobustness measure:

wi(qqq) = �min RRRi +RRRTi : (21)


Fig. 3. Optimal positioning problem of base and pallets.

Fig. 4. DOF manipulator with 1-DOF base.

TABLE ILINK PARAMETERS OF SIMULATED SYSTEM

Fig. 5. Proposed robustness measure.

This measure directly represents the robustness of the closed-loop sta-bility for any given configuration of the manipulator based on the pas-sivity. Recall that the transfer functionPPP (s; qqq

d) is given by themechan-

ical system and the transpose of the Jacobian. Hence, this measure doesnot depend on the underlying control law.

The characteristics are that wi � 0 holds, in general, and wi = 0holds only at the RAC for the ith mode. The configuration where wi

Fig. 6. RAC.

Fig. 7. In-phase regions and boundaries.

Fig. 8. Minimum and nonminimum phase regions.

is near zero is preferable from the viewpoint of the robust stability ofthe closed-loop system. In addition, it is estimated that the system isminimum phase in the neighborhood of the RAC. This point will bediscussed in the next example section.The advantages of this measure are as follows. First, it is easily cal-

culated by n�n numerical computation of eigenvalues. Second, it clar-ifies the optimality of the arm configuration based on the positive real-ness. Third, it does not depend on the underlying control law. Fourth,it identifies the robustness for individual mode.


Fig. 9. Disposition of pallets. (a) Case 1. (b) Case 2.

B. Possible Region of RAC

In general, the computation of the proposed measure wi(qqqd), i =1; � � � ;m over a whole working range is time consuming, especially ifthe system hasmanyDOFs (resulting in havingmany vibrationmodes).In this subsection, an efficient method to find the RAC will be givenby examining the possible region of the RAC.

The possible region is obtained by considering the condition 2 ofthe theorem, i.e., the necessary condition of the positive realness. Ashas been described in the Remark, Section II-D, condition 2 of thetheorem is equivalent to the condition that all diagonal elements ofPPP (s) are in-phase. The in-phase region of each diagonal element canbe easily obtained, since it is surrounded by the uncontrollable and un-observable boundaries [15]. Letting PPP (s) = [Pkl]; 1 � k; l � n,CCC = col[c1; � � � ; cn], and BBB = [bbb1 � � � bbbn], we have Pjj(qqqd) =cccj(sIII � AAA)�1bbbj , j = 1; � � � ; n. Since

�ijj = �2< ��i cccjuuu2(n+i)�1 vvv

�

2(n+i)�1bbbj (22)

the unobservable boundary iLjuo and the uncontrollable boundary

iLjuc

of the Pjj element of the ith mode are obtained by the set of qqqd, givenas

iLjuo = qqqd : vvv

�

2(n+i)�1bbbj = 0 (23)iLjuc = qqqd : cccjuuu2(n+i)�1 = 0 : (24)

Note that the Pjj element becomes uncontrollable on iLjuc and unob-

servable on iLjuo. �

ijj = 0 holds on these boundaries, and the sign

of �ijj changes when the boundary is passed [15]. It is enough to ex-amine wi(qqqd) only in this region to find the RAC, and not necessary toexamine the other region. It contributes to the reduction of the compu-tation time.

The subscript i is omitted when m = 1, then (23) and (24) can bedescribed as

Ljuo = qqqd : rank MMM

j

C < 2N (25)

Ljuc = qqqd : rank MMM

j

O < 2N (26)

where

MMMj

C(qqqd)�= [bbbj AbAbAbj � � � AAA

2N�1bbbj ] (27)

MMMj

O(qqqd)�=col[cccj ; cccjAAA; � � � ; cccjAAA

2N�1]: (28)

Examining the possible region also gives meaningful informationof the characteristic of the RAC. As will be described in the examplesection, it is implied that the RAC is independent to the stiffness andviscosity of the base flexibility.

IV. NUMERICAL EXAMPLE: APPLICATION

TO FLEXIBLE ASSEMBLY SYSTEM

A. Optimal Positioning Problem of the Pallets

The optimal position of the workpiece pallet xwork, the parts-feederpallet xparts, and the base ybase in Fig. 3 is examined. From the viewpoint of the RAC, the two pallets should be placed in the neighborhoodof the RAC where the local asymptotic stability of the original non-linear system is guaranteed for high-bandwidth positioning (settling).Fig. 4 shows the model of the manipulator. The flexibility of the

base is assumed as a passive viscoelastic joint after positioning the ballscrew, but no additional sensor is attached to measure this flexibility.The task given to the system is the following. First, the end-effectormoves from the workpiece to the parts feeder during the time interval[0, 0.1] s, then settles on the parts feeder for [0.1, 0.3] s, moves back tothe workpiece for [0.3, 0.4] s, and finally settles on the workpiece for[0.4 0.6] s.The task is switching between two controllers; a local task-space

controller for the settling task and a simple joint-level tracking con-troller for the transferring task. A simple joint-level tracking control,a joint proportional derivative (PD) control in this example, is appliedfor the transferring task from the parts feeder to the workpiece and itsreverse, since this task is not the main aim of this paper. A straight tra-jectory is designed between the pallets, then joint trajectories are calcu-lated by an inverse kinematics, assuming that the manipulator is rigid.In general, tracking error is unavoidable to some extent, because theflexibility is not compensated, however, the instability problem for thistransferring task is not caused because the collocation holds. A moresophisticated path planning [19] can be applied, if necessary. Note thatxwork = xparts should be satisfied for the shortest transferring be-tween two pallets, since the distance between the tracks is set constant(0.5 m).

B. Dynamic Equations

Table I shows the link parameters of the manipulator shown in Fig. 4.�i is the joint angle of axis i, mi is the mass, li and Ii are the lengthand the moment of inertia of link i, and lgi is the distance between theaxis i and the center of mass. The base flexibility is approximated asa viscoelastic joint with the stiffness and viscous constants given bykp1 = 3:0 � 104 Nm/rad, kv1 = 10 Nms/rad. The working range is�� < �2 < �, 0 < �3 < �. We assume that no singular configu-ration is passed while working. Within the working range, the naturalfrequency of the base vibration is from 70 to 100 Hz.The dynamics equation around qqqd is given as [20]

0

�2

�3

=MMM

��1��2��3

+DDD

_�1_�2_�3

+KKK

�1 � ��1�2 � �2d

�3 � �3d

(29)


Fig. 10. Positioning error. (a) Case 1. (b) Case 2.

Fig. 11. Overview of experimental system.

whereMMM = [Mij ],DDD = diag(kv ; 0; 0),KKK = diag(kp; 0; 0), and

M11 =m1l2

g1 +m2 l2

1 + l2

g2 + 2l1lg2C2

+m3 l2

1 + l2

2 + l2

g3 + 2l1l2C2 + 2l1lg3C23 + 2l2lg3C3

+ I1 + I2 + I3 (30)

M12 =M21 = m2 l2

g2 + l1lg2C2 + I2 + I3

+m3 l2

2 + l2

g3 + l1l2C2 + 2l2lg3C3 + l1lg3C23 (31)

M13 =M31 = m3 l2

g3 + l2lg3C3 + l1lg3C23 + I3 (32)

M22 =m2l2

g2 +m3 l2

2 + l2

g3 + 2l2lg3C3 + I2 + I3 (33)

Fig. 12. Mechanical components of experimental system. (a) Tendon-drivenjoint. (b) Coupling. (c) Infrared LED on tip. (d) Position-sensitive detector.

TABLE IILINK PARAMETERS OF EXPERIMENTAL SYSTEM

Fig. 13. Proposed measure and experimental configuration.

M23 =M32 = m3 l2

g3 + l2lg3C3 + I3 (34)

M33 =m3l2

g3 + I3 (35)

JJJ =�l1S1 � l2S12 � l3S123 �l2S12 � l3S123 �l3S123

l1C1 + l2C12 + l3C123 l2C12 + l3C123 l3C123

(36)

JJJv =�l2S12 � l3S123 �l3S123

l2C12 + l3C123 l3C123

: (37)


Fig. 14. Experimental results A–F (stable configurations). (a) Tip position A(0.0, 0.1). (b) Tip position B(0.15, 0.15). (c) Tip position C(0.3, 0.15). (d) Tip positionD(0.43, 0.0). (e) Tip position E(0.35, 0.2). (f) Tip position F(0.15, 0.25).

Notations such as Sa = sin(�a) or Cab = cos(�a + �b) are used fortrigonometrical functions to simplify the description. For the systemunder consideration, PPP (s) is 2� 2 andm = 1.

The uncontrollable boundaries L1

uc, L2

uc, and the unobservableboundaries L1

uo, L2

uo for P11 and P22 are obtained analytically byapplying (25) and (26)

L1

uc : J12(M13M23 �M12M33) + J13(M12M23 �M13M22) = 0

(38)

L2

uc : J22(M13M23 �M12M33) + J23(M12M23 �M13M22) = 0

(39)

L1

uo : J11 �M2

23 +M22M33 + J12(M13M23 �M12M33)

+ J13(�M13M22 +M12M23) = 0 (40)

L2

uo : J21 �M2

23 +M22M33 + J22(M13M23 �M12M33)

+ J23(�M13M22 +M12M23) = 0 (41)

where Jkl is the (k; l) element of JJJ . This result clearly shows that theboundaries are independent from the stiffness and viscosity of the base.Therefore, the RAC is expected to be independent from the magnitudeof these factors. This independence from the magnitude of the stiffnessand viscous factors is implied only for this simple example, at the mo-ment. However, this characteristic seems to be important, and the detailanalysis will be discussed in our future work.

C. Robustness Evaluation of Arm Configuration

Fig. 5 shows the proposed measure w1(qqqd). A high robustness isobtained when the configuration is determined by w1 as an evaluationfunction. Fig. 6 shows the RACwith the contour plot ofw1(qqqd), where


Fig. 15. Experimental results G–J (instable configurations). (a) Tip position G( 0.07, 0.2). (b) Tip position H(0.3, 0.25). (c) Tip position I(0.3, 0.05).(d) Tip position J(0.25, 0.15).

the distance of the contour line is 0.015. The hatched regions of Fig. 7show the regions for which condition 2 of the theorem is satisfied.L1

uc,L2

uc, L1

uo, and L2

uo are also shown by various dashed or dotted lines.Adjacent regions come in contact with each other at the point wherethe mode is uncontrollable and unobservable.

Fig. 8 shows the minimum phase and nonminimum phase regions.In control theory, it has been said that the minimum-phase propertyis favorable, since it has no instable zeros. As revealed by comparingFig. 6 and Fig. 8, the robustness and the minimum phase property areclosely related.

D. Simulation

The applied settling controller is a task-space PD feedback controllerwithout considering the base flexibilityKKKc(s) = KKKp+KKKvs, then theactuator torque is calculated as

��a = �JJJTv (qqqa)(KKKpyyy +KKKv _yyy) (42)

whereKKKp = diag(1:0� 105; 1:0� 105) N/m, andKKKv = diag(1:5�103; 1:5� 103) Ns/m, respectively. Note that only qqqa and yyy are avail-able in this system, and this control law stabilizes the rigid body mode.To make clear the fundamental control performance for a specific armconfiguration, the control structure and its gains are fixed.

Let us consider two cases, shown in Fig. 9. In case 1, both palletsare placed on the RAC. Fig. 10 shows the positioning error. Let theadmissible error be 100 �m. The error converges satisfactorily within100 ms in case 1, while it oscillates in case 2 and becomes unstable,since the robustness at (0.33, �0.25) is not high enough.

As a result, the validity of optimizing the positions of the pallets hasbeen shown from the viewpoint of the RAC.

V. EXPERIMENT

A. Experimental System

The experimental system is shown in Fig. 11. Its link mechanism isthe same as Fig. 4; a 2-DOF planar manipulator mounted on a 1-DOFflexible base. Two DC motors are attached to each joint. Each motoris equipped with an encoder whose resolution is 2000 pulses per rev-olution. As shown in Fig. 12(a), each arm is tendon driven, in order toprevent backlash of gear-driven systems. The gear ratio is 1:5.2. Thebase is supported by two bearings, and the shaft is connected to theground through a coupling, as shown in Fig. 12(b). This coupling playsthe role of the elastic joint, since it has a rotational stiffness. The linkparameters are shown in Table II.As shown in Fig. 12(c), an infrared light-emitting diode (LED) is

attached at the tip of the arm. The position of the tip is measured by atwo-dimensional position-sensitive detector (PSD) shown in Fig. 12(d).In this system, C2044 of Hamamatsu photonics is used whose sensiblearea is 4:7� 4:7 mm. A PC/AT machine with a Pentium III 200-MHzprocessor is used as a controller. The position of the tip and the pulse ofthe encoder are sent to an AD/DA interface board installed in the PC.The sampling time is 25 �s.

B. Arm Configuration and Controller

Ten configurations are selected for the experiment, as shown inFig. 13: A(0.0, 0.1), B(0.15, 0.15), C(0.3, 0.15), D(0.43, 0.0), E(0.35,�0.2), F(0.15, �0.25), G(�0.07, 0.2), H(0.3, 0.25), I(0.3, 0.05),and J(0.25, �0.15) as the tip position. The proposed measure w iscalculated for the experimental system. It is superimposed to Fig. 13,where the distance of the contour line is 0.15. Among the selectedconfigurations, A–F are almost on the RAC, as shown in Fig. 6. On


the other hand, the configurations G–J are off the RAC. It is expectedthat the closed-loop system is stable from A to F, since the robustnessis the highest, while the system might be instable from G to J, sincethe robustness is not high enough.

Equation (42) is used as a controller, whereKKKp = diag(2400:0; 2400:0) N/m, KKKv = diag(25:6; 25:6) Ns/m,respectively. In this experiment, the fixed gains are applied for eachconfiguration.

The positioning-error trajectories are measured when the tip is re-leased from the initial position within the sensible range, so that theinitial error is maximum 1.2 mm. The stability of the system can bejudged sufficiently by this response.

C. Results

Fig. 14 shows the positioning error from A to F, and Fig. 15 showsthe positioning error fromG to J. As revealed in Fig. 14, the closed-loopsystems from A to F are stable, since these configurations are almoston the RAC. On the contrary, as shown in Fig. 15, in the configurationsfrom G to J which are off the RAC, an oscillation is induced and finallydiverged.

The results clearly show the validity of the concept of the RAC. Theexperimental system behaved exactly as expected from the theoreticalanalysis with the proposed measure.

VI. CONCLUSION

In this paper, we defined the RAC of a manipulator mounted on aflexible base where the linearized dynamics is positive real. A localasymptotic stability with a high closed-loop gain is guaranteed for atask-space feedback control by the Lyapunov indirect method and thepassivity theory. As a result, a high positioning performance can beapplied in the neighborhood of the RAC without considering the baseflexibility, namely, an additional sensor or a solution of a whole inversedynamics is not necessary. Considering the positive semidefinitenessof the residue matrices, we proposed a measure which measures thedistance from the RAC. Moreover, we showed that an in-phase designapproach cannot be applied to MIMO systems.

The validity of the proposed approach was confirmed by a numer-ical example of the flexible assembly system. The positions of the baseand the pallets were optimized for a high-speed positioning with alocal task-space controller. Experiments were performedwith an exper-imental system consisting of a 2-DOF planar manipulator and a 1-DOFflexible base.

REFERENCES

[1] S. Arimoto and F. Miyazaki, “Asymptotic stability of feedback laws forrobot manipulators,” in Proc. 1st IFAC Symp. Robot Control, 1985, pp.447–452.

[2] R. H. Cannon and E. Schmitz, “Initial experiments on the end-point con-trol of a flexible one-link robot,” Int. J. Robot. Res., vol. 3, no. 3, pp.62–74, 1984.

[3] M. W. Spong, “Modeling and control of elastic joint robots,” J. Dynam.Syst., Meas., Contr., vol. 109, pp. 310–319, 1987.

[4] P. Tomei, “A simple PD controller for robots with elastic joints,” IEEETrans. Automat. Contr., vol. AC-36, pp. 1208–1213, Oct. 1985.

[5] K. Osuka and F. Matsuno, “On robustness of passivity of manipulators,”in Proc. IEEE Conf. Decision and Control, 1999, pp. 3406–3409.

[6] A. De Luca, “Feedforward/feedback laws for the control of flexiblerobots,” in Proc. IEEE Int. Conf. Robotics and Automation, 2000, pp.233–240.

[7] J. Onoda and R. T. Haftka, “An approach to structure/control simulta-neous optimization for large flexible spacecraft,” AIAA J., vol. 25, no. 8,pp. 1133–1138, 1987.

[8] A. M. A. Hamdan and A. H. Nayfeh, “Measures of modal controllabilityand observability for first- and second-order linear systems,” Trans.AIAA, J. Guid. Contr., vol. 12, no. 3, pp. 421–428, 1989.

[9] D. S. Bodden and J. L. Junkins, “Eigenvalue optimization algorithms forstructure/controller design iterations,” Trans. AIAA, J. Guid. Contr., vol.8, no. 6, pp. 697–706, 1985.

[10] S.M. Joshi, “Control of large flexible space structures,” in Lecture Notesin Control and Information Sciences. New York: Springer-Verlag,1989.

[11] K. Ono and T. Teramoto, “Design methodology to stabilize the naturalmodes of vibration of a swing-arm positioning mechanism,” ASME Adv.Inform. Storage Syst., vol. 4, pp. 343–359, 1992.

[12] S. Hara, “Integration of structure design and control system synthesis,”in Proc. TITech COE/SMS Symp., 1998, pp. 53–60.

[13] T. Iwasaki, “Integrated system design by separation,” in Proc. IEEEConf. Control Application, Aug. 1999, pp. 97–102.

[14] T. Iwasaki, S. Hara, and H. Yamauchi, “Structure/control integrationwith finite frequency positive real property,” in Proc. TITech COE/SuperMechano-System Symp., 2000, pp. 126–135.

[15] T. Yoshikawa and J. Ueda, “Task priority-based mode-shaping methodfor in-phase design of flexible structures aiming at high speed and ac-curate positioning,” in Proc. IEEE Int. Conf. Robotics and Automation,2001, pp. 1806–1812.

[16] J. Ueda and T. Yoshikawa, “Robust arm configuration of manipulatormounted on flexible base,” in Proc. IEEE Int. Conf. Robotics and Au-tomation, 2002, pp. 1321–1326.

[17] C. T. Chen and C. A. Desser, “Controllability and observability of com-posite systems,” IEEE Trans. Automat. Contr., vol. AC-12, pp. 402–409,Apr. 1967.

[18] R. Bellman, Introduction of Matrix Analysis. New York: McGraw-Hill, 1970.

[19] D. N. Nenchev, K. Yoshida, P. Vichitkulsawat, and M. Uchiyama, “Re-action null-space control of flexible structure mounted manipulator sys-tems,” IEEE Trans. Robot. Automat., vol. 15, pp. 1011–1023, Dec. 1999.

[20] T. Yoshikawa, Foundations of Robotics. Cambridge, MA: MIT Press,1990.

Documents

Robust Arm Configuration of Manipulator Mounted on Flexible Base