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Engineering, 2010, 2, 118-123 doi:10.4236/eng.2010.22017 Published Online February 2010 (http://www.scirp.org/journal/eng). Copyright © 2010 SciRes. ENGINEERING A Set of Globally Stable N-PID Regulators for Robotic Manipulators Baishun Liu, Fancai Lin, Bingli Tian Department of Battle & Command, Academy of Naval Submarine, Qingdao, China E-mail: baishunliu@163.com, lfcai_777@yahoo.com.cn, woshixiaobenmiao@126.com Received August 28, 2009; revised September 28, 2009; accepted October 4, 2009 Abstract This paper deals with the position control of robot manipulators with uncertain and varying-time payload. Proposed is a set of novel N-PID regulators consisting of a linear combination of the proportional control mode, derivative control mode, nonlinear control mode shaped by a nonlinear function of position errors, linear integral control mode driven by differential feedback, and nonlinear integral control mode driven by a nonlinear function of position errors. By using Lyapunov’s direct method and LaSalle’s invariance principle, the simple explicit conditions on the regulator gains to ensure global asymptotic stability are provided. The theoretical analysis and simulation results show that: an attractive feature of our scheme is that N-PID regu- lators with asymptotic stable integral actions have the faster convergence, better flexibility and stronger ro- bustness with respect to uncertain and varying-time payload, and then the optimum response can be achieved by a set of control parameters in the whole control domain, even under the case that the payload is changed abruptly. Keywords: Manipulators, Robot Control, PID Control, N-PID Control, Global Stability 1. Introduction It is well known that PID controllers can effectively deal with nonlinearity and uncertainties of dynamics, and asymptotic stability is achieved accordingly [1–3]. Hence, most robots employed in industrial operations are con- trolled through PID controllers that introduce integral action by integrating the error. It is well known that inte- gral-action controllers with this class of integrator often suffer a serious loss of performance due to integrator windup, which occurs when the actuators in the control loop saturate. Actuator saturation not only deteriorates the control performance, causing large overshoot and long settling time, but also can lead to instability, since the feedback loop is broken for such saturation. To avoid this drawback, various PID-like controllers have been proposed to improve the transient performance. For ex- ample, PID controllers consisting of a saturated-P, and differential feedback plus a PI controller driven by a bounded nonlinear function of position errors [4], a lin- ear PD feedback plus an integral action of a nonlinear function of position errors [5], a linear derivative feed- back plus a PI term driven by a nonlinear function of position errors [1], a linear PD feedback plus double in- tegral action driven by the positions error and the filtered position [6], a linear PD feedback plus an integral action driven by PD controller [7], and a linear PD feedback plus an integral action driven by NP-D controller [8], are presented recently. In this paper, we propose a set of new global position controllers for robots which do not include their dynam- ics in the control laws. Motivated by the idea that is in- troducing the nonlinear action of the position errors into PD-NI controller [5] and modifying the nonlinear inte- gral action via injection of the required damping so that the transient performance of the closed-loop system may be improved, we develop a set of new N-PID-like regu- lators consisting of a linear combination of the propor- tional control mode, derivative control mode, nonlinear control mode shaped by a nonlinear function of position errors, linear integral control mode driven by differential feedback, and nonlinear integral control mode driven by a nonlinear function of position errors. The simple ex- plicit conditions on the regulator gains to ensure global asymptotic stability are provided. Throughout this paper, we use the notation )(A m and )(A M to indicate the smallest and largest eigen- values, respectively, of a symmetric positive define bounded matrix)(xA , for any. The norm of vec- tor n Rx x is defined asx T xx , and that of matrix A B. S. LIU ET AL.119 is defined as the corresponding induced norm AAA T M . The remainder of the paper is organized as follows. Section 2 summarizes the robot model and its main properties. Our main results are presented in Sections 3 and 4, where we briefly review some known PID-like control laws and present a set of new N-PID-like control laws, and then provide the conditions on the controller gains to ensure global asymptotic stability, respectively. Simulation examples are given in Section 5. Conclusions are presented in Section 6. 2. Problem Formulation The dynamic system of an n-link rigid robot manipulator system [1] can be written as: uqgqDqqqCqqM )(),()( (1) where is the vector of joint positions, is the vector of applied joint torques, is the symmetric positive define inertial matrix, is the vector of the Coriolis and centrifugal torques, is the positive define diagonal fric- tion matrix, and is the vector of gravita- tional torques obtained as the gradient of the robot po- tential energy due to gravity. q n 1n n (qg )(q u ) C( 1n nn (qM qqq ), 1 Dn ) 1n U A list of properties [1] of the robot dynamic model (1) is recalled as follows: )()()(0 MqMM Mm (2) 0),(2)( qqCqM T (3) n R 22),(0 qCqqqCqC Mm (4) n Rqq , where and are all positive constants. m CM C For the purpose of this paper, it is convenient to in- troduce the following definition and properties [5]. Definition 1: ),,( x F with01 , 0 , and denotes the set of all continuous differential in- creasing functions, n Rx T n xfxfxf )](,),([)( 1such that |||)(||| xxfx |:| xRx |)(| xf |:| xRx 0)()/(1 xfdxd (5) where stands for the absolute value. || Figure 1 depicts the region allowed for functions be- longing to set),,( xF . For instance, the tangent hyper- bolic function belongs to set and the Arimoto sine function, whose entries are given as follows: ),1),1(tanh( xF 2/1 2/||)sin( 2/1 )( xif xifx xif xf (6) which belongs to set . ),1),1(sin( xF The important properties of function belonging to set )(xf ),,( xF are now established. The function satisfies for all, )(xfx Tn Rx 0)( xfxT (7) The Euclidean norm of satisfies for all )(xf n Rx , 22 ||||||)(|| xxf (8) nxf ||)(|| (9) Throughout this paper, we use the notation d qqq P KD K , to indicate the position errors, is the desired joint position, which is assumed to be constant; , , , , , and are all positive define diagonal d q S KPf K n I K n IP KID K matrices. 3. N-PID-Like Control Laws 3.1. PID-Like Control Law Review To put our contribution in perspective, we will briefly review some known PID-like control laws, as follows: 1) Semiglobally stable PID control law [1], t IDP dqKqKqKu 0 )( . 2) Globally stable PD-NI control law [1], t IDP dqKqKqKu 0 ))(tanh( . where )tanh( is the hyperbolic tangent vector function. 3) Globally stable PD-NPI control law [1], t ISDPdqsKqsKqKqKu 0 ))(()( . Figure 1. ),,( xF functions. Copyright © 2010 SciRes. ENGINEERING B. S. LIU ET AL. Copyright © 2010 SciRes. ENGINEERING 120 where is the differential function of a class of ap- proximate potential energy function. )(s 4) Semiglobally stable PI2D control law [6], t IDP dqqKqKqKu 0 )]()([ . 5) Semiglobally stable PD-IPD control law [7], t IDIPDPdqKqKqKqKu 0 )]()([ . 6) Globally stable PD-INP-D control law [8], t IDIP DP dqKqK qKqKu 0 )]())(tanh([ . 7) Globally stable SP-D-NPI control law [4], t bI bPfDP dqfK qfKqKqsatKu 0 ))(( )()( . where is the normal saturated function and is a bounded nonlinear function. )(sat )( b f 8) Globally stable PD-NI control law [5], t IDP dqfKqKqKu 0 ))(( (10) where is a continuous differential increasing bounded functions shown in Figure 1. )(f Most of the present industrial robots are controlled through PID-like controllers above. Although these con- trollers have been shown in practice to be effective for position control of robot manipulators, unfortunately most of them often suffer a serious loss of performance, that is, causes large overshoot and long settling time due to unlimited integral action. Based on the above fact, we get intuitively an idea that the transient performance of the closed-loop system may be improved if the nonlinear action of position errors is intro- duced into the control law (10) and the damping is injected into its integrator. Following this idea, a set of novel N-PID-like regulators is developed in the next subsection. 3.2. Our N-PID Control Laws The new nonlinear PID control laws are proposed as fol- lows, t IDIP PfDP dqKqfK qfKqKqKu 0 )]())(([ )( (11) It is worthy to note that the control law above is con- sisting of a linear combination of the proportional control mode, derivative control mode, nonlinear control mode, linear integral control mode and nonlinear integral control mode. Hence, based on the five control modes above, five differential N-PID-like control laws with the same stability as the control law (11) can be derived, as follows: t IDIPDP dqKqfKqKqKu 0 )]())(([ (12) t IDIP PfD dqKqfK qfKqKu 0 )]())(([ )( (13) t IDIPD dqKqfKqKu 0 )]())(([ (14) t IPDPdqfKqKqKu 0 ))(( (15) t IPPfDP dqfKqKqKqKu 0 ))(()( (16) Discussion 1: It is obvious that the control law (11) can be simplified to other PID-like control laws such as P-NI control, NPI control, PD-INP-D control [8], and so on. Moreover, the control law (15) is the same as the one (10) reported by [5]. This shows that the control law (11) is a natural extension of them and implies that its application is not limited in the robots, too. Discussion 2: Notice that the integral actions in con- trol laws (10) and (11) can be rewritten as )(qfKI andqKqfK IDIP )( , respec- tively. From this, it is easy to see that the latter has the same stability as the one presented by [7,8], that is, they are all asymptotically stable but the former has the same stability as the classical integral action, qKI IP , that is, they are all only stable. This means that the con- troller (11)-(14) should have faster convergence, better flexibility than the one (10), and then the controller (11)-(14) can yield higher performance of control, too. Moreover, integrator windup can be avoided by choosing suitable parameters K andID K . Discussion 3: Compared to the classical PID con- trol and PD-NI control [5], the following observations can be made during the control process: when 0)( qKqfK IDIP, the integral action remains con- stant; if the integral action is large, increases, and then the integral action instantly decreases, vice versa. However, the integral action produced by the classical or nonlinear integrator [5] always increases as long as the error does not cross over zero, only when the error crosses over zero, the integral action will start to de- crease. This shows that the control laws with the asymp- totically stable integrator should have the faster conver- gence and better flexibility, and then can yield higher transient performance, once again. q 4. Stability Analysis For analyzing the stability of the closed-loop system, it is convenient to introduce the following notation. Defining , )]0()([))(()(1 0 qKqgKdqftz IDdIP t B. S. LIU ET AL.121 and then the control law (11) can be rewritten as, )()( )( dIPPf DIDP qgzKqfK qKqKKu (17) The closed-loop system dynamics is obtained by sub- stituting the control action from (17) into the robot dynamic model (1), u 0)()( )()()(),()( qfKzKqKK qgqgqDKqqqCqqM PfIPIDP dD (18) From the closed-loop system dynamics above, it is easy to see that the origin nTTTTR z qq 3 0),,( is the unique equilibrium point. Now, the objective is to provide conditions on the con- troller gains, , P KD KPf K , IP K and ID K ensuring global asymptotic stability of the unique equilibrium point. This is established in the following. Theorem: Consider the robot dynamics (1) together with control law (11). There exists positive constant small enough, and choose the gain matrices, , , and such that a DP K K Pf KIP KID K IMKKK MIPIDP)(4 (19) 2 ||||)( 4 1 )()()( qaqKKKq qgqqUqU IPIDP T d T d (20) 2 ||)(|| ))(( )()()]()()[( qfa qKKKqf qfKqfqgqgqf IPIDP T Pf T d T (21) InCMDK MMD )( (22) hold, and then the closed-loop system (18) is globally asymptotically stable, i.e. 0li m q t. Proof: To carry out the stability analysis, we consider the following Lyapunov function candidate: n i q iiDiPfii T IPIDP T T d T d IP TT idxDKKxf qKKKq qqMqfqgqqUqU qzKqzqqMqV 10 ))(( )( 2 1 )()()()()( )()( 2 1 )( 2 1 (23) where ; , and are the diagonal element of the matrices , and , respec- tively. ii qx Pfi KDi K K i D D K Pf D 1) Positive definition of Lyapunov function candidate. Now, considering the following inequality, and using (8), we have, )())(4)(( 4 1 )( 4 1 )()()( )](2)[()](2[ 4 1 )()( )( 4 1 )( 4 1 qfIMKKKqf qKKKq qfqMqf qfqqMqfq qqMqf qKKKqqqMq MIPIDP T IPIDP T T T T IPIDP TT (24) Substituting (20) and (24) into (23), and using (7) and (19), for any, we obtain, 0),,( TTTT zqq 0)()( 2 1 ))(( ||||)( 4 1 10 2 qzKqz dxDKKxf qaqqMqV IP T n i q iiDiPfii T T i (25) This shows that the Lyapunov function candidate (23) is positive define. 2) Time derivative of Lyapunov function candidate. The time derivative of Lyapunov function candidate (23) along the trajectories of the closed-loop system (18) is, )()())(( )()()( )()()()( )())(()( 2 1 )( qzKqzqDKKqf qKKKqqgqqgq qqMqfqqMqf qqMqqfqqMqqqMqV IP T DPf T IPIDP T d TT TT TTT (26) Substituting )( qfz , and from (18) into (26), and using (3), we have, qqM )( )()( ))(( )]()()[(),()( )())(()( qfKqf qKKKqf qgqgqfqqqCqf qqMqqfqDKqV Pf T IPIDP T d TT T D T (27) Now, using (2) and (5), we get, 2 ||||)()())(( qMqqMqqfM T (28) By using (4) and (9), we obtain, 2 ||||),()( qnCqqqCqfM TT (29) Incorporating (21), (28) and (29) into (27), we obtain, 2 ||)(|| ))(( qfa qInCIMDKqV MMD T (30) From (22), (30) and, we can conclude. 0a0V Copyright © 2010 SciRes. ENGINEERING B. S. LIU ET AL. Copyright © 2010 SciRes. ENGINEERING 122 rge and , in th he n Using the fact that the Lyapunov function candidate (23) is a positive define function and its time derivative is a negative semi define function, we conclude that the equilibrium point of the closed-loop system (18) is stable. In fact, means and. By invoking the LaSalle’s invariance principle, it is easy to know that the equilibrium point is globally asymp- totically stable, i.e., . 0V 0q ),(TT qq 0 q 0q 0 T lim t Remark 1: For the control laws (12)-(16), the globally asymptotically stable results can be derived under some similar sufficient conditions presented by (19)-(22). The proof can follow the similar argument and procedure. It is omitted because of the limited space. Discussion 4: From the proof procedure above, it is easy to see that: 1) if we chooses gain matrix, ID K la enough, the linear term, qKPthe nonlinear term, fKPf e control law (11) is not necessary for guaranteeing global asymptotic stability of the closed- loop system and this means that the global asymptotic stability can still be ensured by the simplified form of the control law (11); 2) on the other hand, with the linear term, qKP, and tlinear term, )( qfKPf , on )( q , it is at the condition (19)-(21) is more easily satis- fied and this results in that the control engineers have more freedom to choose the controller parameters, and then make them more easily tune a high performance controller. obvious th 5. Simulations To illustrate the effect of the controller given in this pa- per, two-link manipulators shown in Figure 2 are consid- ered. The dynamics (1) is of the following form [9] 222112 2 111222112 112112 2 211212111 2 2 uGqqFqFqMqM uGqqFqFqMqM where: , )cos(2)(2212 2 22 2 12111 qllmlmlmmM , , 2 2222 lmM )cos( 2212 2 2212qllmlmM )sin(22121211 qllmFF , )sin()sin()( 212211211 qqglmqglmmG )sin( 21222qqglmG . The normal parameter values of the system are se- lected as:, , 2. kgmm 1 21 mll1 21 /10 smg The desired (set point) positions are chosen as: when , , st 101 1 d q1 2 d q; when , , ; sts 2010 3 1 d q2 2 d q when , . st200 21 dd qq The simulation is implemented by using the control laws (11) and (14), respectively. In simulation, Arimoto Figure 2. The two-link robot manipulators. sine function (6) is used as the nonlinear function of the control laws. The gain matrices of the control law (11) are selected as: )310,310(diagKP , , )150,150(diagKD )500,500( diagKIP , , )200,200(diagKID and )100,100(diagKPf . The gain matrices of the control law (14) are given as: )30,60(diagKD , , )500,1000(diagKIP and )600,1250(diagKID . The simulations with sampling period of 2ms are im- plemented. Figures 3 and 5 present the response of the robot manipulators under the normal parameters. Figures 4 and 6 are the simulation results under the case that the mass is substituted for when kgm1 2 st 20 kgm 3 2 s10 , corresponding to moving payload of 2kg. From the simulation results, it is easy to see that: 1) The optimum response can all be achieved, respec- tively, by the control laws (11) and (14) with a set of control parameters in the whole domain of interest, even under the case that the payload is changed abruptly; 2) These tow controllers used in simulation all have the faster convergence, better flexibility and stronger robustness with respect to uncertain payload, which means that the controllers (12) and (13) should have the same high performance of control as the controllers (11) and (14) because they all employ the same integrators; 3) Comparing the Figures 3 and 5, the controller (11) is easier to achieve the control of high speed and high performance than the controller (14) because the former has two freedom parameters. Figure 3. Under normal parameters, the simulation results with controller (11). B. S. LIU ET AL. Copyright © 2010 SciRes. ENGINEERING 123 exact knowledge of the payload. An attractive feature of our scheme is that the control laws with the asymptoti- cally stable integrators have the faster convergence, bet- ter flexibility and stronger robustness with respect to uncertain and varying-time payload, and then the opti- mum response can be achieved. The explicit conditions on the regulator gains to ensure global asymptotic stabil- ity of the overall closed-loop system are given in terms of some information exacted from the robot dynamics. Our findings have been corroborated numerically on a two DOF vertical robot manipulators. Figure 4. Under perturbed parameters, corresponding to moving payload, the simulation results with controller (11). 7 . References [1] Y. X. Su, Nonlinear Control Theory for Robot Manipula- tors, Science Publishing, Beijing, 2008. [2] J. Alvarez-Ramirez, I. Gervantes, and R. Kelly, “PID regulation of robot manipulators: Stability and perform- ance,” System & Control Letters, Vol. 41, No. 2, pp. 73–83, April 2000. [3] A. A. Pervozvanski and L. B. Freidovich, “Robust stabil- ity of robotic Manipulators by PID controllers,” Dynamics and Control, Vol. 9, No. 3, pp. 203–222, September 1999. [4] S. Arimoto, “A class of quasi-natural potentials and Hy- per-stable PID servo-loops for nonlinear robotic system,” Transactions of the Society of Instrument and Control En- gineers, Vol. 30, No. 9, pp. 1005–1012, September 1994. Figure 5. Under normal parameters, the simulation results with controller (14). [5] R. Kelly, “Global positioning of robotic manipulators via PD control plus a class of nonlinear integral actions,” IEEE Transactions on Automatic Control, Vol. 43, No. 7, pp. 934–938, July 1998. [6] R. Ortega, A. Loria, and R. Kelly, “A semiglobally stable output feedback PI2D regulator for robot manipulators,” IEEE Transactions on Automatic Control, Vol. 40, No. 8, pp. 1432–1436, August 1995. [7] B. S. Liu and F. C. Lin, “A semiglobally stable PD-IPD regulator for robot manipulators,” to be published in 2009 International Conference on Measuring Technology and Mechatronics Automation proceeding. Figure 6. Under perturbed parameters, corresponding to moving payload, the simulation results with controller (14). [8] B. S. Liu and B. L. Tian, “A globally stable PD-INP-D regulator for robot manipulators,” to be published in 2009 International Conference on Information Technology and Computer Science proceedings. 6. Conclusions [9] S. J. Yu, X. D. Qi, and J. H. Wu, Iterative Learning Con- trol Theory & Application, Machine Publishing, Beijing, 2005. In this paper, we have presented a set of solution to the problem of set point regulation for rigid robots without |