MARCH 2009 VOLUME 17 NUMBER 2 IETTE2 (ISSN 1063-6536)

PAPERS

A Bottom-Following Preview Controller for Autonomous Underwater Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Silvestre, R. Cunha, N. Paulino, and A. Pascoal 257

Sensorless Tilt Compensation for a Three-Axis Optical Pickup Using a Sliding-Mode Controller Equipped With aSliding-Mode Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. C.-P. Chao and C.-Y. Shen 267

Humidity and Pressure Regulation in a PEM Fuel Cell Using a Gain-Scheduled Static Feedback Controller . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Y. Karnik, J. Sun, A. G. Stefanopoulou, and J. H. Buckland 283

Nonconservative Robust Control: Optimized and Constrained Sensitivity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.-M. Fransson, T. Wik, B. Lennartson, M. Saunders, and P.-O. Gutman 298

Optimal Real-Time Scheduling of Control Tasks With State Feedback Resource Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. El Mongi Ben Gaid, A. S. Çela, and Y. Hamam 309

Speed Observer and Reduced Nonlinear Model for Sensorless Control of Induction Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. K. Khalil, E. G. Strangas, and S. Jurkovic 327

Adaptive Control of Electrostatic Microactuators With Bidirectional Drive . . . . . . . . . . K. P. Tee, S. S. Ge, and F. E. H. Tay 340Identification of a Managed River Reach by a Bayesian Approach . . . . . .. . . . . . M. Thomassin, T. Bastogne, and A. Richard 353Online References Reshaping and Control Reallocation for Nonlinear Fault Tolerant Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Benosman and K.-Y. Lum 366Optimal Control of a Variable Geometry Turbocharged Diesel Engine Using Neural Networks: Applications on the ETC

Test Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Omran, R. Younes, and J.-C. Champoussin 380

(Contents Continued on Back Cover)

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(Contents Continued from Front Cover)

BRIEF PAPERS

Nonlinear Control Allocation for Non-Minimum Phase Systems . . . . . . M. Benosman, F. Liao, K.-Y. Lum, and J. L. Wang 394Coordination Planning: Applying Control Synthesis Methods for a Class of Distributed Agents . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. T. Seow, M. T. Pham, C. Ma, and M. Yokoo 405Active Joint Synchronization Control for a 2-DOF Redundantly Actuated Parallel Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Shang, S. Cong, Y. Zhang, and Y. Liang 416Velocity Estimation: Assessing the Performance of Non-Model-Based Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Bascetta, G. Magnani, and P. Rocco 424Feedforward Control of Shape Memory Alloy Actuators Using Fuzzy-Based Inverse Preisach Model . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. K. Nguyen and K. K. Ahn 434Vision-Based Range Regulation of a Leader-Follower Formation . . . . P. Vela, A. Betser, J. Malcolm, and A. Tannenbaum 442Moving Horizon Tracking Control of Wheeled Mobile Robots With Actuator Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Chen, M.-M. Ma, H. Wang, Z.-Y. Liu, and Z.-X. Cai 449An Adaptive Controller for the Shunt Active Filter Considering a Dynamic Load and the Line Impedance . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. A. Valdez, G. Escobar, and R. Ortega 458Analog Active Control of Acoustic Noise at a Virtual Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Pawelczyk 465A Systematic Method for Gain Selection of Robust PID Control for Nonlinear Plants of Second-Order Controller

Canonical Form .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. H. Chang and J. H. Jung 473Decentralized Detection of a Class of Non-Abrupt Faults With Application to Formations of Unmanned Airships . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Léchevin and C. A. Rabbath 484Noniterative -Based Model Order Reduction of LTI Systems Using LMIs . . . . . . . . . . . . . . . . . A. Nobakhti and H. Wang 494

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416 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 17, NO. 2, MARCH 2009

Active Joint Synchronization Control for a 2-DOFRedundantly Actuated Parallel Manipulator

Weiwei Shang, Shuang Cong, Yaoxin Zhang, and Yanyang Liang

Abstract—This brief applies the synchronization to solve con-trol problem of redundantly actuated parallel manipulators. Withthe synchronization method, a new controller termed active joint-synchronization (AJ-S) controller is developed for a 2-degree-of-freedom (DOF) redundantly actuated parallel manipulator. Thedynamic model of the parallel manipulator is formulated in theactive joint space, in which the internal force is calculated by theprojection method and the friction is depicted with the Coulomb� viscous friction model. By defining the tracking error, synchro-nization error, coupled error, and the referenced trajectory vectorof the active joints, the AJ-S controller based on the dynamic modelis designed. And the AJ-S controller is proven to guarantee asymp-totic convergence to zero of both tracking and synchronization er-rors by the Barbalat’s lemma. The AJ-S controller is implementedin the trajectory tracking experiments of an actual 2-DOF redun-dantly actuated parallel manipulator, and the superiority of theAJ-S controller over the well-known tracking controller is studied.

Index Terms—Friction model, joint space, parallel manipulator,synchronization, tracking controller.

I. INTRODUCTION

P ARALLEL manipulators have the advantages of low in-ertia, high stiffness and high overload driven capability,

and so on. These advantages avoid the drawbacks of serial ma-nipulators and make parallel manipulators be potential high per-formance motion platforms [1]. In order to realize the potentialhigh performance motion, it is very important to design con-trollers. Current controllers for parallel manipulators can be di-vided into two categories. The first kind is kinematic controller,including proportional-derivative (PD) controller [2], nonlinearPD (NPD) controller [3], and artificial-intelligence-based con-troller [4], etc. The nonlinear dynamics of parallel manipulatorsis not considered in these controllers, so the complex computa-tion of dynamics can be avoided and the controller design canbe simplified greatly. However, the performances of these con-trollers are always limited, and the stabilities of these controllersat high-speed are not guaranteed. The second kind is dynamiccontroller. Unlike the kinematic controller, full dynamic modelof the parallel manipulator is taken into account in dynamic con-troller. So the nonlinear dynamics of the parallel manipulatorscan be compensated and higher performance can be achieved.

Manuscript received February 08, 2008. Manuscript received in final formMay 15, 2008. First published November 18, 2008; current version publishedFebruary 25, 2009. Recommended by Associate Editor C.-Y. Su. This work wassupported by the National Natural Science Foundation of China under Grant50375148.

The authors are with the Department of Automation, University of Sci-ence and Technology of China, Hefei, Anhui 230027, P. R. China (e-mail:wwshang@mail.ustc.edu.cn; scong@ustc.edu.cn; yxzhang3@mail.ustc.edu.cn;sunnylg@mail.ustc.edu.cn).

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.2008.2000978

The traditional dynamic controllers include the augmented PD(APD) controller and computed-torque (CT) controller [5], [6].Recently, some novel controllers with performance indicatorswere presented for parallel manipulators [7]–[9]. These novelcontrollers can improve the performances of the traditional dy-namic controllers further, but also require more accurate dy-namic model. However, it is very difficult to get the accuratevalues of the dynamic parameters of parallel manipulators. Boththe kinematic controllers and the dynamic controllers are in-troduced directly from the controllers of the serial manipula-tors. For the serial manipulators, satisfactory performances canbe obtained with these controllers. But, when these controllersare applied to the parallel manipulators, the mechanism struc-ture characteristics of a number of branch chains of the par-allel manipulators are not taken into account. From the view-point of controller design, the controller of an active joint ineach branch cannot respond to other active joints in the differentbranch chains. Thus, the correlation among the branch chains isignored, and the motion accuracy of the parallel manipulators athigh-speed is poor.

Therefore, it is necessary to analyze the characteristics of themechanism structure of parallel manipulators, and develop acontrol strategy according to those characteristics. On the as-sumption that the branch chains of the parallel manipulator werecut at the common point, the parallel manipulator can be trans-formed into a new system of multi serial manipulators, whichare constrained by the internal force. Considering the correla-tion of the multi serial manipulators, we can realize the syn-chronization motion of the parallel manipulator. In recent years,synchronization with the cross-coupled control technology hasattracted many researchers in motion control field, as it can getcoordination motion of complex systems. Koren [10] presentedcross-coupled control first, then Tomizuka et al. [11] studiedthe synchronization of two motion axes with the cross-coupledcontrol. Nijmeijer et al. [12], [13] analyzed the synchronizationcontrol theory of generalized dynamic systems, and summarizedsome basic ideas of the synchronization control. In recent years,synchronization control has been applied to complex mechan-ical system by many researchers. Sun et al. [14], [15] realizedposition synchronization of multiple motion axes with the cross-coupled control. Kang and Yeh [16] utilized the synchronizationfor attitude control of multi-satellite systems. Synchronizationcontrol also has been used for robot systems, such as coordina-tion manipulation of multi robotic manipulators [17]–[19], for-mations switching for swarms of mobile robots [20], real-timegrasping force optimization of multi-fingered dexterous hands[21], etc. For parallel manipulators, Su et al. [22] combined thecross-coupled control with the simple NPD controller, and pro-posed a synchronization controller for set-point position reg-ulation of a planar parallel manipulator. Then this controllerwas improved further by incorporating the proportional-integral(PI) algorithm [23]. In order to realize position synchronization

1063-6536/$25.00 © 2008 IEEE

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SHANG et al.: ACTIVE JOINT SYNCHRONIZATION CONTROL FOR A 2-DOF REDUNDANTLY ACTUATED PARALLEL MANIPULATOR 417

control in trajectory tracking, the dynamics of the parallel ma-nipulator must be considered in the synchronization controller.Based on the dynamic model, a new synchronization controlleris developed for a planar parallel manipulator [24], [25].

The main contribution of this brief is the application of thesynchronization to solve control problem of redundantly actu-ated parallel manipulators. With the synchronization method, anew controller termed active joint-synchronization (AJ-S) con-troller is developed for a 2-degree-of-freedom (DOF) redun-dantly actuated parallel manipulator. In order to design the AJ-Scontroller, the dynamic model is established in the active jointspace for the parallel manipulator. In fact, the established modelis equivalent to a constrained system with multi serial manipula-tors, and the constraint is depicted with the internal force amongthe branch chains of the parallel manipulator. Also, in the dy-namic model, the friction of three active joints is described withthe Coulomb viscous friction model which are neglected in[24], [25]. According to the synchronization control idea, thesynchronization error and coupled error vector are defined. Withthe defined synchronization and coupled error vector, the de-sired trajectory vector is changed into the referenced trajectoryvector to increase the tracking accuracy. Using the error vectorsand the referenced trajectory vector, the AJ-S controller basedon the dynamic model is designed in the active joint space. Withthe AJ-S controller, the internal force among the branch chainsand the friction of the three active joints can be compensated.Thus, the synchronization ability between the branch chainsand trajectory tracking accuracy of the end-effector can be im-proved. The AJ-S controller is proven to guarantee asymptoticconvergence to zero of both tracking and synchronization errors.The AJ-S controller is implemented in the trajectory trackingexperiments of an actual 2-DOF redundantly actuated parallelmanipulator. The experiment results show that, the proposedAJ-S controller can get satisfactory trajectory tracking accuracyof the end-effector and synchronization relation among the ac-tive joints.

This brief is organized as follows. In Section II, the dynamicmodel of a 2-DOF redundantly actuated parallel manipulator isestablished in the active joint space. In Section III, the AJ-S con-troller is developed, and it is proven to guarantee asymptoticconvergence to zero of both position and synchronization errors.In Section IV, the tracking controller is designed for the parallelmanipulator to compare with the AJ-S controller. In Section V,the trajectory tracking experiment of an actual 2-DOF redun-dantly actuated parallel manipulator is carried out, and the ex-periment results are compared with the tracking controller. InSection VI, several important remarks are concluded.

II. DYNAMIC MODEL IN THE ACTIVE JOINT SPACE

In order to design the AJ-S controller, we will formulate thedynamic model of a 2-DOF redundantly actuated parallel ma-nipulator in the active joint space first. As shown in Fig. 1, areference frame is established in the workspace for the parallelmanipulator. The unit of the frame is meter. The parallel ma-nipulator is actuated by three servo motors located at the baseA1–A3, and the end-effector is mounted at the common joint

, where the three chains meet. Coordinates of the three basesare A1 (0,0.25), A2 (0.433,0), and A3 (0.433,0.5), and all of thelinks have the same length 0.244 m.

Fig. 1. Coordinates of the 2-DOF redundantly actuated parallel manipulator.

In Fig. 1, represents the th active joint whose coordinate isdenoted by , represents the th passive jointwhose coordinate is denoted by , representsthe end-effector whose coordinate is denoted by .Here, subscript 1,2,3 are adopted to represent the th serialmanipulator. Furthermore, let be the angels of the activejoints, be the angels of the passive joints, and be the lengthof the links. From Fig. 1, one can obtain

(1)

Differentiating (1), and solving the joint angles yields

(2)Expressing (2) with the matrix form as

(3)

where is the active joint vector,is the passive joint vector, is the velocity Jaco-

bian matrix between the active joints and the end-effector, isthe velocity Jacobian matrix between the passive joints and theend-effecor. From (3), the velocity relation between the activejoints and passive joints can be written as

(4)

where . Let , and (4)can be briefly expressed as

(5)

Differentiating (5), the acceleration relation between the activejoints and passive joints can be written as

(6)

Cutting the parallel manipulator at the common point inFig. 1, one can have an open-chain system including three inde-pendent 2-DOF serial manipulators, each of which contains anactive joint and a passive joint. The dynamic model of the par-allel manipulator equals to the model of the open-chain systemunder the closed-loop constraints, thus, the dynamics of the par-allel manipulator can be formulated by combining the dynamicsof the three serial manipulators under the constraints.

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418 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 17, NO. 2, MARCH 2009

According to [26], the dynamic model of each 2-DOF serialmanipulator can be formulated as

(7)

where ; is inertia matrix; is Coriolis andcentrifugal force matrix; is joint torque vector,where is the active joint torque, the passive joint torque

; is friction torque vector, where the passivejoint friction torque can be ignored and only the active jointsfriction torque is considered. The active joint friction torque

can be formulated as

(8)

where represents the Coulomb friction, and representsthe coefficient of the viscous friction.

Combine the dynamic models of the three 2-DOF serial ma-nipulators, and unite the active joint equation and the passivejoint equation, respectively, then the dynamic model of theopen-chain system can be expressed as

(9)

where and are thefriction torque vector of the active joints and passive joints, re-spectively; and are theforce vector of active joints and passive joints respectively;is inertia matrix; is Coriolis and centrifugal force matrix.

Substituting (5) and (6) into (9), the dynamic model of theopen-chain system in the active joint space can be written as

(10)

Let , then (10) can be expressed as

(11)

Based on (11) and the internal forces due to the closed-loopconstraints, the dynamic model in the active joint space of theparallel manipulator can be expressed as

(12)

In (12), represents the internal force vector, where ma-trix is the differential of the closed-loop constrained equationand is a multiplier representing the magnitude of the internalforces. Considering and , then and

. Multiplying both sides of (12) by , one can ob-tain

(13)

Then, (13) can be briefly expressed as

(14)

where and .

In the dynamic model (14), is an unknown multiplier rep-resenting the magnitude of the internal forces, and it is difficultto be measured directly. Fortunately, the unknown multipliercan be calculated by linear projection method [27]. Multiplyingboth sides of (14) by , one can obtain

(15)Solving the above equation yields

(16)

III. ACTIVE JOINT SYNCHRONIZATION CONTROL

In the active joint synchronization control, an additionalfeedback signal termed the synchronization error is introduced.The synchronization error represents the coordination relationamong the active joints and is not equivalent to the conventionaltracking error. For the parallel manipulator, the synchronizationerror of each active joint contains the information from itselfand the adjacent joints. Hence, the motions of all active jointsare coordinated and the tracking accuracy of the end-effector issubstantially smaller than that without using the synchroniza-tion error. The motivation of defining synchronization error ispresented in details in [25].

Let be the desired trajectory of the th active joint, thenthe tracking error can be defined as

(17)

and the tracking error vector can be written as

(18)

With the synchronization proposed in [14], we would like tokeep the tracking error as , and regulatethe coordination relation among the three active joints to keepfollowing equation satisfied:

(19)

The above equation can be divided into three sub-goals as

(20)According to the concept of synchronization, the synchroniza-tion errors between all possible pairs of two neighboring activejoints can be defined as

(21)

where denotes the synchronization error of the active joint .One can find that, the synchronization relation (20) of the threeactive joints can be satisfied when . Unlike the tradi-tional non-synchronized control considering the position con-trol only, here the synchronization control between two neigh-boring active joints is also required so that (20) holds. Then thesynchronization error vector can be written as

(22)

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SHANG et al.: ACTIVE JOINT SYNCHRONIZATION CONTROL FOR A 2-DOF REDUNDANTLY ACTUATED PARALLEL MANIPULATOR 419

According to the cross-coupled control idea proposed in [14],the coupled error includes both the tracking error andsynchronization error , which can be defined as

(23)

where coupled parameter is a positive constant. Define thedeviation vector of the adjacent synchronization errors as

(24)

then the coupled error (23) can be rewritten as

(25)

where is coupled parameter matrix. Then thecoupled velocity error vector can be written as

(26)

With the coupled error (25) and coupled velocity error (26),the combined error can be defined as

(27)

where combined error parameter is a positive constant. Ac-cording to (27), represents a linear differential equationwhose unique solution is . So the convergence problemof and can be reduced to keep zero. (27) can berewritten with the vector expression as

(28)

where is parameter matrix of the combinederror and the velocity combined error vector can be written as

(29)

The referenced tracking velocity and acceleration can be de-fined as

(30)

(31)

Based on the dynamic model (14), the AJ-S control law canbe designed as

(32)

where and are both symmetric, positive definite matricesof constant gains. The control law (32) can be separated into

five parts according to their different functions. The first partis the dynamics compensation, which can be written as

. The second part is the friction compensation,which can be written as . The third part is the internalforce compensation, which can be written as .The fourth part is the coupled error elimination, which can bewritten as . The fifth part is the synchronizationerror elimination, which can be written as . Thus,the control law (32) can be rewritten as

(33)

In the following, we will prove the asymptotic stability of theparallel manipulator system controlled by the AJ-S controller.First, we will introduce an assumption and two lemmas [28].

Assumption 1: and it’s first- and second-order derivativesare bounded.

Lemma 1: If the differentiable function is bounded anddecreasing , then it converges to a limit.

Lemma 2 (Barbalat): If the differentiable function has afinite limit as , and if is uniformly continuous, then

as .Theorem 1: The proposed AJ-S controller (32) guarantees

asymptotic convergence to zero both of tracking error andsynchronization error of the system (14), i.e., and

as .Proof: Choose Lyapunov function candidate as

(34)

Differentiating with respect to time yields

(35)The closed-loop system equation can be written as

(36)

Multiplying both sides of the above equation by , and thensubstituting the resulting equation into (35) yields

(37)

Considering the structural properties, then one can haveand

(38)

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420 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 17, NO. 2, MARCH 2009

where

(39)

With (23) and (27), one can obtain

(40)

Substituting (40) into (39) yields

(41)

Substituting (41) into (38) yields

(42)

Therefore, the system is global stability, and , , arebounded. From the definitions of and , one can knowis bounded. Next, we will prove the convergence of andwith the Lemma 2.

Now since and , is bounded and de-creasing. With the Lemma 1, converges to a limit. From theexpression (34) of , this implies that , , are bounded.

Furthermore, this in turn implies that and are bounded con-sidering the Assumption 1. And is uniform positive definite,then exists and bounded. Thus, the closed-loop system(36) can be written as

(43)

So is also bounded.Since is bounded, one can find that and are both

bounded according to (28), also is bounded according to (26)and is bounded according to (21) and is bounded accordingto (24). Considering

since , , and are bounded, is bounded. Thus,is uniformly continuous. With the Lemma 2, when ,

, and this implies and as .Then one knows that and as accordingto (28). Since and as , one knowsas from (23).

IV. TRACKING CONTROLLER

As the performance comparison with the AJ-S controller, thewell-known tracking controller proposed by Slotine and Li [28]is designed for the parallel manipulator.

Let be the vector of the desired trajectory of the activejoints, the actual tracking error can be defined as

(44)

Define vector as

(45)

where is symmetric, positive definite matrix of constant gains.Then the referenced tracking velocity can be defined as

(46)

Based on the dynamic model (14), the control law of the trackingcontroller can be designed as

(47)

where is symmetric, positive definite matrix of constantgains. The control law (47) can be separated into four parts ac-cording to their different functions. The first part is the dynamicscompensation, which can be written as . Thesecond part is the friction compensation, which can be writtenas . The third part is the internal force compensation,which can be written as . The fourth part is thetracking error elimination, which can be written as .

The tracking control law (47) and the AJ-S control law (32)shared common parts including friction compensation andinternal force compensation . So the tracking con-troller also need calculate the internal force when it’s used forthe parallel manipulator. In order to realize the internal forcecompensation in the two control laws, the unknownmagnitude of the internal forces must be calculated with (16).From (16), one can see, in order to get the magnitude , the

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SHANG et al.: ACTIVE JOINT SYNCHRONIZATION CONTROL FOR A 2-DOF REDUNDANTLY ACTUATED PARALLEL MANIPULATOR 421

Fig. 2. Prototype of the 2-DOF redundantly actuated parallel manipulator.

actual velocity signal and acceleration signal should beknown. We will discuss how to get and in Section V.

The biggest difference of the two control laws is the errorelimination part. The tracking control law (47) only contains thetracking error elimination. But the AJ-S control law (32) con-tains two parts of the coupled error and synchronization errorelimination, which can eliminate the tracking error and synchro-nization error simultaneously.

V. EXPERIMENTS

As shown in Fig. 2, the actual experiment platform is a2-DOF redundantly actuated parallel manipulator produced byGoogol Tech Ltd., Shenzhen, China. The parallel manipulatoris equipped with three permanent magnet synchronous servodrives with harmonic gear drives. The active joint anglesare measured with the absolute optical-electrical encoders,and the joints are controlled by the motion control board ofGT-400-PCI-SV from Googol Tech Ltd. We realized the AJ-Scontroller and tracking controller with Visual C++. In theactual experiment, the control algorithms ran on a Pentium IIICPU at 733 MHz, and sampling period is 2 ms. The desiredtrajectory of the end-effector is a straight line, the starting pointis (0.22,0.19) and the ending point is (0.35,0.29), thus, themotion distance is 0.164 m. The profile of the desired velocityis a trapezoidal curve [29]. The maximum velocity is 0.2 m/sand the maximum acceleration is 5 m/s .

In order to realize the internal force compensation ,the actual velocity signal and acceleration signal should beknown. But, in practice, it is not easy to get acceleration signalif only the joint angles can be measured. In our actual experi-ment, the velocity signals of the joints are got by numerical dif-ferentiation of the joint angles measured with the optical-elec-trical encoders and the acceleration signals of the joints are gotby numerical differentiation of the velocity signals. Due to thetwice differentiation, the acceleration signals are quite noisy, soa low-pass filter is adopted to filter the velocity signal before thecalculation of acceleration [30].

The dynamic and friction parameters used in the two con-trollers can be found in [30]. In fact, other control parametersin the two controllers are tuned and determined by the actual

Fig. 3. Tacking errors of the three active joints: (a) joint 1; (b) joint 2;(c) joint 3.

experiments. In the experiment, the tracking controller param-eters are tuned as follows: ,

. The AJ-S controller parameters are tunedas follow: the coupled parameter matrix , thecombined error parameter matrix , thecombined error gain matrix , and the syn-chronization error gain matrix .

The experiment results of the trajectory tracking are shown inFigs. 3 and 4. The tracking error , , and of the threeactive joints are shown in Fig. 3(a)–(c), respectively. From thethree figures, one can see that the tracking accuracy of the threeactive joints is improved obviously with the AJ-S controller.Fig. 4(a)–(c) are the synchronization error , respec-tively. One can see that three synchronization errors are smaller

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422 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 17, NO. 2, MARCH 2009

Fig. 4. Synchronization errors among the three active joints: (a) between joint1 and joint 2; (b) between joint 2 and joint 3; (c) between joint 3 and joint 1.

with the AJ-S controller, thus, the coordination of the three ac-tive joints are better with the AJ-S controller.

Furthermore, to evaluate the performances of the two con-trollers, the root-square mean error (RSME) are selected as per-formance index. The tracking RSME (T-RSME) of the activejoints can be defined as follows:

(48)

where represents the tracking error at the th samplingpoint of the th active joint, which is defined by (17), and

TABLE IRSMEs OF THE TWO CONTROLLERS

is the number of the sample point. Also, the synchronizationRSME (S-RSME) of the active joints can be defined as follows:

(49)

where represents the synchronization error at the th sam-pling point of the th active joint, which is defined by (21).

The T-RSME and S-RSME results are shown in Table I, inwhich the RPE means the reduced percentage of error. FromTable I, one can see that the T-RSME decreases 19.1%, and theS-RSME decrease 34.2% using the proposed AJ-S controller,compared with the tracking controller.

VI. CONCLUSION

A new synchronization controller termed AJ-S controller isdeveloped for a 2-DOF redundantly actuated parallel manip-ulator. The AJ-S controller is proved to guarantee asymptoticconvergence to zero both of the tracking error and synchroniza-tion error. According to the best knowledge of the author, this isthe first time that the synchronization control was applied to a re-dundantly actuated parallel manipulator. Our theoretic analysisimplies that, the proposed controller can improve the coordi-nation between the adjacent joints, and ultimately increase thetracking accuracy of the end-effector. Our experiment resultsshow that, compared with the tracking controller, the trackingerror and the synchronization error of the three active joints aredecreased obviously with the proposed AJ-S controller.

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