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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 18, NO. 1, JANUARY 2010 143

Dynamic Compensation Control of Flexible MacroMicro Manipulator Systems

Tang Wen Yang, Member, IEEE, Wei Liang Xu, Senior Member, IEEE, and Jian Da Han, Member, IEEE

AbstractMacromicro architecture, which consists of macroand micro manipulators, is used here to eliminate errors at the tip

of a flexible manipulator. The macro uses long arms and has suchadvantages as larger work volume and lower energy consumptionbut suffers from large deformations and vibrations. The micro is asmaller rigid manipulator and is attached on the end of the macroto isolate the system endpoint from the undesirable flexibility ofthe macro. Using perturbation theories, a new kinematical methodis introduced, first, by redefining the micros motion as a means ofcompensating for the errors at the endpoint of the macro. Then, anexcellent practical control scheme is proposed to realize the end-point control with the feedback of joint angles and vibrations. APD controller is applied to the micro, which augmented the com-pensation quantities. To damp out vibrations, a nonlinear controllaw is proposed for the macro, taking the interacting dynamicsof the micro to the macro into account. The compensation andcontrol algorithms work very well on a macromicro setup, andnumerous experimental results prove the applicability of the pro-posed schemes.

Index TermsEndpoint control, error compensation, flexiblemanipulators, macromicro systems.

I. INTRODUCTION

LARGE LIGHT robotic systems now feature in many space

missions [1]. Not only are they playing a more impor-

tant role in space-station construction and extravehicular-ac-

tivity support but they can also be used to transfer payloads, re-

place orbital units, and maintain the elements of space station.Besides, large robotic systems find applications in the fields

of aircraft cleaning [2] and nuclear waste clear up [3], where

long-reach operation is required. Usually, a large robotic system

has long and slender arms. For example, the developing Cana-

dian Mobile Servicing System in the International Space Sta-

tion is approximately 17 m long when all the arms are fully ex-

tended. Such a robotic system is far from being stiff and is often

described as a flexible arm. Compared with heavy and bulky

industrial robots, flexible arms have such obvious advantages

as larger work volume and lower energy consumption, but they

Manuscript received February 27, 2008. Manuscript received in final formOctober 31, 2008. First published April 14, 2009; current version publishedDecember 23, 2009. Recommended by Associate Editor R. Moheimani. Thiswork was supported in part by the National Science Fund of China underGrant 60305008, by the State Key Laboratory of Robotics, CAS under GrantRL200702, and by the Beijing Jiaotong University under Grant 2007XM007.

T. W. Yang is with the School of Computer and Information Technology,Beijing Jiaotong University, Beijing 100044, China (e-mail: [email protected]).

W. L. Xu is with the School of Engineering and Advanced Tech-nology, Massey University, Auckland 102 904, New Zealand (e-mail:[email protected]).

J. D. Han is with the Shenyang Institute of Automation, Shenyang 110016,China (e-mail: [email protected]).

Color versions of one or more of the figures in this brief are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCST.2008.2009529

suffer large deformations and low-frequency vibrations, typi-

cally caused by structural flexibility. Consequently, issues such

as motion planning and dynamic modeling become very com-

plicated, and end-effector position and force control are even

more challenging.

Over the past decades, these issues have received intensive at-

tention, with no generic solution to date. Widely used modeling

methods include the assumed-mode method and the finite-el-

ement method with either Lagrangian or NewtonEuler recur-

sive formulations. On the control side, a variety of strategies,

like singular perturbation methods [4] and Lyapunov-based con-

trollers [5], have been investigated. Vibration is one of the crit-

ical problems to control a flexible manipulator, and many ap-

proaches to suppress vibrations have been reported, such as

input shaping techniques [6]. Recently, wave-based strategies[7] are proposed to absorb the vibration energy inside a flexible

system. Macromicro architecture has also been introduced as

a way to improve the motion performance of flexible manipula-

tors while suppressing the vibrations. It consists of a large macro

and small micro manipulators and, thus, combines the merits of

large and small manipulators. A large macro manipulator has

large workspace but more or less limits the dexterity and speed

at its endpoint, while a small micro one is attached on the end

of the macro, providing fast precise motion at the tip point.

Macromicro manipulator systems are structurally stable and

well suitable for fast and precise endpoint positioning. More-

over, the lower inertia design of a micro manipulator is helpfulfor precise force control [8]. Generally, a flexible macromicro

manipulator system is redundant and dynamically nonlinear,

and the structural flexibility and the dynamic coupling between

the macro and the micro make the control issue much compli-

cated. Control-law design for the system is not an easy job and

very challenging.

The idea of mounting a smaller manipulator on the end of

a flexible manipulator was introduced initially in [9]. A quick

wrist, namely, micro part, is attached to a flexible manipulator,

forming a flexible macromicro manipulator system. Research

on the micro control mainly focuses on reducing the effects of

the macro flexibility. In [3], a motion compensator is added to

the industrial controller of a flexible macromicro system, using

strain-gauge signals. In [10], a command filter is added again to

the previous controller as an input prefilter. George and Book

[11] made use of the interacting dynamics between the micro

and the macro to damp the macro vibration. In [12], the control

gain matrices of the micro are carefully designed with a fre-

quency-matching method, with no elastic-state measurements

needed. As a macromicro system is redundant, motion plan-

ning of such a system is a tough job, and the endpoint control

becomes tremendously difficult. Zhanget al.[13] used the EDA

algorithm with Gaussian probability model to generate optimal

joint motions of a macromicro system. Yoshikawa et al.[14] in-

troduced compensability and compensability-measure concepts

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144 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 18, NO. 1, JANUARY 2010

in motion planning, and a quasi-static trajectory control law is

proposed to track the endpoint of a flexible macromicro ma-

nipulator system. In [15], a dynamic trajectory tracking con-

trol is discussed, taking the system dynamics into account. With

endpoint-position sensing, Ballhaus and Rock [16] proposed an

endpoint control scheme. In [17], nonlinear inversion and pre-

dictive control laws are designed for the micro and the macro, re-spectively. Recently, neural networks [18] and fuzzy logic [19]

were also applied to the control of macromicro manipulator

systems. The problem of force control was addressed in [20], as

the micro contacts the environment.

In this brief, endpoint control of a flexible macromicro ma-

nipulator system is discussed on the base of an error-compensa-

tion method and a feedback control law. The errors at the end-

point are compensated for through the fast and precise motions

of the micro. Using perturbation theories, the error-compensa-

tion method is introduced first in the next section. Then, a new

and practical control scheme is proposed with the feedback of

the joint motions and vibrations in Section III. Section IV gives

the experimental results on a planar flexible macromicro ma-nipulator system to attest the proposed methods. Finally, con-

clusions are drawn out in Section V.

II. ERRORCOMPENSATION OF AFLEXIBLEMACROMICRO

MANIPULATORSYSTEM

The macro manipulator deploys the micro to the vicinity of

a work site, where the dexterity of the micro can be then used

to perform specific operations. Simultaneously, the errors at the

tip of the macro are eliminated through the precise motion of

the micro. In many researches published, the micro joint mo-

tion is resolved into coarse and fine components. The coarsecomponent is planneda prioriby assuming a rigid kinematics.

The fine motion commits to compensating for the errors at the

system endpoint and is computed by complicatedJacobians, as

done in [14] and [21]. In [23], the errors at the tip of a flex-

ible manipulator are reduced significantly through the actuator

extra motion, obtained by perturbation theories. The work is ex-

tended here to flexible macromicro manipulator systems. Al-

ternatively, we attempt to eliminate the errors at the endpoint of

a flexible macromicro manipulator by the micro, not the macro

extra motions. Besides, the micro motions are not resolved into

two parts.

A. Tip Errors of a Flexible Macro Manipulator

A flexible macromicro manipulator system is shown in

Fig. 1. The macro is a large manipulator with degrees of

freedom, and the micro is a smaller one with degrees of

freedom. The coordinate system is established first. Each of

the coordinate frame is located at either end of a link. The

solid lines illustrate the flexible macromicro system with the

macro links bending. If no bending takes place, the system then

becomes a rigid macromicro system, as shown in the figure

with the dashed lines. is the base frame.

represents the frame of the th link of the rigid system and is

displaced to , due to the elastic deformations of the

link. As the macro deformations are accumulated from the first

to the th link, the tip frame of the macro

Fig. 1. Flexible macromicro manipulator system.

moves to . is the endpoint frame of

the macromicro manipulator system.

As aforementioned, the displacements of each frame lead to

a shift at the tip frame and are increasingly ac-

cumulated at the endpoint of the macromicro system even-

tually, which deteriorate an operation accuracy. The displace-

ments come from many aspects, such as link and joint flexibili-

ties, mechanical inaccuracies, etc. Compared with the length of

a flexible link, the displacements are very tiny, allowing us to

take them as perturbations in [22] and [23]. Therefore, pertur-

bation theories are used here to derive the total errors at the tip

of the macro first.Usually, the deformations of the flexible links contribute most

of the displacements. Then, the displacements due to the link de-

formations are our concern in this brief. But without loss of gen-

erality, now, let us set the displacements of the frame

of the th link to be

(1)

where is the translation displacements and

is the angular displacements, in the frame

.

To calculate the total errors at the tip frame of the macro,

the displacements of every frame are transformed into

, assuming no displacements from the other

frames. From perturbation theories [22], the displacements at

the tip of the macro are

(2)

where is theJacobianmatrix of the displacement vector of

to .


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YANGet al.: DYNAMIC COMPENSATION CONTROL OF FLEXIBLE MACROMICRO MANIPULATOR SYSTEMS 145

Now, the total errors can be obtained by summing up all the

displacements as

(3)

where is the total translation errorvector and is the total rotation errors

of , with respect to .

B. Compensating Motion of the Micro Manipulator

is expressed in and, undoubtedly, leads

to a shift at , if the micro are not adjusted. In order to

keep the position and orientation of unchanged, the

shift should be eliminated by adjusting some joint motions. A

kinematical approach is presented here by redefining the micro

motions for the purpose.

Define and as the transformation matrices of

and to , respec-

tively, the transformation of the two can be then established

with

(4)

(5)

where are the joint angles of the macro and

is the transformation matrix of any

two adjacent coordinate frames.

Instead, the position and orientation errors at the tip of the

macro can be seen as the cause of a transformation of

to the current axes of . Then,we have

(6)

Substituting (4) into (6), we obtain

(7)

where is a transformation caused by the total translation and

rotation error vector , i.e.,

(8)

Set to be the transformation of to

, and if no compensation with the micro,

we have

(9)

where are the desired joint motions of

the micro and planneda priori.

As can be seen, macromicro architecture has a big advantage

over traditional robots to eliminate the shift without adjusting

all the joints of a system. Different from [21] and [22], in this

brief, we use the derived to redefine the joint motions of

the micro so as to eliminate errors at the endpoint. Going back

to Fig. 1, we can see that can be interpreted alternatively

Fig. 2. Plannar flexible macromicro system.

as the combination of a series of homogenous transformations

of , and

rewritten as

(10)

where are the redefined joint motions

of the micro.

Then, from (9) and (10), we obtain

(11)

The link deformations of the macro can be measured in

real time with a sensing system, which will be described in

Section IV, and the micro desired motion is planned a priori.

Then, in (11), its left term is known. Now, the problem becomes

finding the micro joint variables , andessentially, it is a problem of inverse kinematics (more details to

the inverse kinematics problem are out of the scope of this brief

and can be found in [24]). For some cases, it is not necessary to

plan the micro joint angles. That is to say, the micro is regulated

to a configuration with all the joints fixed initially, and then is

exclusively used to compensate for the endpoint errors.

C. Simulation Case

As an example of demonstration, a flexible macromicro ma-

nipulator system shown in Fig. 2 is used to verify the com-

pensation approach. The physical parameters of the simulation

system are taken from an experimental system, which will beintroduced in Section IV. Both the macro and the micro have

2 DOF, and all the joints are revolute and perpendicular to the

motion plane. Therefore, the displacements at the tip frame of

the macro are the position errors in the - and -axes and the

orientation error of . To entirely compensate for the errors at

the tip of the macro, the micro should have the same number or

more degrees of freedom than a task space. As the micro dis-

cussed here has 2 DOF, only the position errors of the macro

are to be compensated for, and the orientation error is ignored

here. However, the approach can be easily applied to any flex-

ible spatial macromicro system.

In the simulation, all the joints are planneda priorito follow

sine trajectories, and the first two dominant modes of the two

flexible links are presumed to be excited and to be calculated


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Fig. 3 Position errors at the endpoint of the macro.

from the link deformations of the macro deliberately. In prac-tice, the link deformations are measured in real time with the

sensing system introduced in Section IV. Fig. 3 shows the end-

point-position errors in the base frame. They are computed from

the link deformations and the joint angles of the macro. Fig. 4

shows the micro motions. The dashed lines are the desired an-

gles while the solid lines the redefined motions which dedicate

to eliminate all the position errors at the end of the system.

III. CONTROLLERDESIGN OF A MACROMICRO

MANIPULATORSYSTEM

A macromicro manipulator system has a large arm carrying

on a smaller manipulator. The macro provides the system a large

workspace, whereas the micro has a limited workspace but can

move faster and is more dexterous. Its endpoint reflects the be-

havior of the flexible macromicro system. In other words, the

micro isolates the endpoint from the undesirable flexibility of

the macro. But inevitably, there exists dynamic interaction be-

tween the macro and the micro. In the following, we discuss the

endpoint-control problem by taking the interaction into account.

The equations of motion of a flexible macromicro manipu-

lator system can be derived by using Lagrangian formulations

and have the form of

(12)

Fig. 4. Desired and redefined joint angles of the micro.

where and are the joint variables of

the macro and the micro. are the deformations of the

macro flexible links. is the system inertia, and

. is the nonlinear centrifugal,

elastic, and Coriolis force. is the system stiffness matrix.

and are the drive torques/forces of the macro and the micro,

respectively.

The micro commits to implement error compensation with

fast response. Although the perturbation at the macro end acts

on the micro, simple and effective control laws are preferred forthe rigid micro in many applications. Herein, an industrial PD

control law is adopted, combining the compensation quantities,

and given by

(13)

where and are the positive proportional and deriva-

tive gain matrices. is the measured joint variables of the

micro, and is the redefined joint angles of the micro,

namely, .

The macro uses long, lightweight, and therefore elastic links

to be capable of performing long-reach operations but suffers

from the undesirable deformations of the elastic links as the

system runs at high speed. If the endpoint is directly chosen as


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the output and the driving torques are chosen as the inputs, the

macro is nonminimum phase. This is recognized as the difficulty

of endpoint control of flexible arms, and the elastic modes must

be constrained so as not to spoil the stability of the whole con-

trol system.

The equations of motion of the flexible macromicro manip-

ulator system are written in a form that is a function of joint vari-ables and link deformations, but here, we control the endpoint

motion of the system. Therefore, the system dynamics model

needs to be transformed into the operational space, namely, the

task space, for the purpose of control design. As is non-

singular matrix, (12) can be then rewritten as

(14)

where and are the and identity matrices,

respectively.

Set to be the system output vector in the task

space, then it can be calculated from the joint variables and the

link deformations, i.e.,

(15)

Differentiating (15) twice, we have

(16)

where is theJacobianof , with respect to the joint

velocities of the macro and the micro and the link-deformation

rates to the endpoint velocities.

Substituting (16) into (14), we obtain

(17)

where

Fig. 5. Block diagram of the endpoint control scheme.

Then, in the task space, a feedback linearization control law

is proposed below for the flexible macro, given by

(18)

where is the inner loop of the control to be determined.is the generalized inverse of . If the number

of degrees of freedom of the macro is equal to that of the task

space , becomes .

Here, the inner loop of the control law has the form of

(19)

where is the endpoint trajectory of the flexible macromicro

system, planned with a rigid system. and are the

positive feedback gain matrices.

Now, (13), (18), and (19) establish the framework of an end-

point control scheme, and Fig. 5 shows the block diagram of thecomplete control system. As the control input vector is spec-

ified in the task space, a planner is used to obtain the joint vari-

ables of the micro while its control is realized in the joint space.

In Section II, (2), (3), (7), and (11) give the expression of the

compensation algorithm. is the desired joint motions of the

micro, namely, in (9). Note that if

is given with the micro joints fixed, then should be replaced

by the feedback of the micros joint position to compute the

redefined variables with (11). For this case, the micro is ex-

clusively dedicated to compensate for the errors at the endpoint

of the macro.

The stability of the control system is analyzed in the taskspace by introducing the system-error dynamics. Let us define

as the tracking errors, it is then bounded. Substituting

(18) and (19) into (17) yields

(20)

Then, the system-error dynamics can be rewritten as a linear

form of

(21)

where


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Fig. 6. Experimental macromicro system.

To prove the control system stability, we define a matrix to be

the solution of the following equations:

(22)

where is a specified positive-definite matrix.

The solution of (22) is symmetric positive definite. Now, a

Lyapunov function candidate can be defined to be

(23)

Obviously, . Differentiating it and substituting

(21) and (22) into the resulting equation, we have

(24)

It can be seen that . According to the invariant-set

theorem [25], the control system is asymptotically stable, and

the system errors tend toward zero. The parameter uncertainties

in (18) was not studied in the stability analysis of the control

scheme, but the experimental results as follows prove the control

stability appropriately.

IV. EXPERIMENTALRESULTS

A planar flexible macromicro system, shown in Fig. 6,

was constructed to do researches on flexible manipulators. The

macro is a 2-DOF flexible-link manipulator. The micro is a

five-bar kinematic chain, with the same number of degrees offreedom as the macro. The cranks of the micro are same in

length, so are the connecting links. All the joints are revolute,

with the axes perpendicular to the horizontal motion plane. The

physical parameters of the flexible macro and the rigid micro

are given, respectively, in Tables I and II. The reach of the

micro is 0.10 m, in comparison with a total reach of 0.90 m of

the macro.

The two links of the macro use lightweight aluminum alloy

with rectangular cross section. The deformations and vibrations

are therefore confined to the motion plane. The joints of the

macro are driven by two servo ac motors with a 25:1 gear re-

duction on the shoulder and 5:1 on the elbow. The micro is ac-

tuated by two identical direct-drive dc motors. Each the joint

of the macromicro system is equipped with an encoder on the

TABLE IPHYSICALPARAMETERS OF THEFLEXIBLEMACRO

TABLE IIKEYPARAMETERS OF THEFIVE-BARMICRO

Fig. 7. Laser diode and PSD.

Fig. 8. Endpoint motion in the Cartesian space.

motor shaft to measure that the angle, and the velocity, is calcu-

lated here from the angle variable using backward difference.

An industrial control computer is used to handle all the real-

time processing. Data communications are done with the con-

trol laboratory cards on the bus. A three-channel quadrature

card is used to count the pulses from the encoders to obtain

joint position. All the control commands are sent to the motors

through card and driveamplifier. The analog signals of the

torque sensors and the position-sensitive detectors (PSDs) are

converted into digital data through . PSD is an optical-sen-

sitive device with fast response and high resolution and applied

to measure deformations and vibrations at the end of each the


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Fig. 9. Responses of the macro joints.

Fig. 10. Position errors at the endpoint with no compensation.

macro flexible link and working with laser diode. The PSD is

mounted on one end of a flexible link, and the laser diode on the

other end of the link, as shown in Fig. 7. Note that we only con-

cern the deformations and vibrations on the motion plane. In the

previous study [22], an optical-sensing system was presented to

Fig. 11. Responses of the micro joints.

measure spatial deformations, and the principle of measurement

was also introduced.

All the real-time software codes are developed inC++. Here,

we attempt to control the endpoint of the macro to move from

an initial point ( m, ) to a target point of

( m, m) in the base frame, following a parabolic

trajectory, given by

(25)

The trajectory is well covered by the reach of the macro, and

the micro exclusively commits to compensate for the errors at

the endpoint of the system. The initial positions of the micros

two joints are 30 and , respectively. As introduced ear-

lier, the micro joint variables are fed back to redefine the next

micro motion. All the controller gains were chosen empirically

to ensure the control stability, and they can be determined with

the estimation approach introduced in [26]. The control gains

for the micro are , , , and

, and the gain matrices for the macro controller are

given respectively by


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Fig. 12. Position errors at the endpoint with the micros compensation.

In Section III, a general control strategy is introduced for a

flexible macromicro system. However, as shown in Fig. 6, the

micro mass is mainly placed on the end of the macro and its size

is much smaller than that of the macro. Here, the micro is pre-

sumed to be a lumped mass to derive the equations of motion of

the macromicro system, and the dynamic interaction between

the macro and the micro is thus equivalent to the lumped mass.

Actually, (18) is the nonlinear feedback control law for the flex-ible macro.

The experimental results are shown in Figs. 8, 9, 10, 11, and

12. Fig. 8 shows the endpoint positions of the system. Fig. 9

gives the joint responses of the macro. As shown, the response

of the second joint, namely, the elbow joint, is trembling at the

beginning of motion due to gravity, since no balance masses

were used to counteract the suspension system. Fig. 10 shows

the errors of the first experiment at the endpoint of the micro in

the Cartesian space, with no compensating motions added, and

they are obtained from the feedback of the macro link deforma-

tions and its joint variables. For comparison, the micro is used

in the second experiment to compensate for the position errors

at the tip by redefining its joint motion, as shown in Fig. 11. The

tip position errors are shown in Fig. 12. It can be seen that the

largest errors are less than 0.25 cm in the -direction and 1.0

cm in the -direction. Although the endpoint errors cannot be

entirely eliminated, they are far less than the results of the first

experiment. Moreover, the errors caused by the residual vibra-

tions of the macro are nearly eliminated by the micro motion

after a tracking maneuver is finished.

V. CONCLUSION

In this brief, the macromicro architecture, consisting of a

large macro and a smaller micro manipulator, is presented to im-

prove the endpoint motion of flexible manipulators. The errors,

caused by link deformations, mechanical or sensing inaccura-

cies, etc., are transformed to the errors at the end of the macro,

and the micro motions are redefined to compensate for the er-

rors with perturbation approaches. A PD controller is chosen for

the micro to meet the demand of fast response, augmented with

the compensating terms. As the inertia of the micro is able to

damp the macro vibrations, a feedback nonlinear control is pro-

posed for the macro, which takes the dynamic interaction of themicro to the macro into account. The approaches proposed are

applied to a flexible macromicro manipulator system, and the

experimental results show that the errors at the system endpoint

are reduced considerably, and the control strategy is stable and

applicable.

ACKNOWLEDGMENT

The authors would like to thank the reviewers who have

helped improve this brief.

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