Robust Tracking Control of Robot Manipulator Using Dissipativity Theory

Transcription

1 Moern Applie Science July 008 Robust racking Control of Robot Manipulator Using Dissipativity heory Hongrui Wang Key Lab of Inustrial Computer Control Engineering of Hebei Province Yanshan University Qinhuangao China Department of Electronic an Informational Engineering Hebei University Baoing 0700 China Zhanfang Feng (Corresponing author) Key Lab of Inustrial Computer Control Engineering of Hebei Province Yanshan University Qinhuangao China Xiuling Liu Department of Electronic an Informational Engineering Hebei University Baoing 0700 China Abstract he robust H controller is esigne for the problem of rigi robot tracking which base on the issipativity theory. he quaratic form issipative feeback control law was given for interference suppression uner the conition of existing moel error an external isturbance. he scheme improve the robustness of the system. he simulation results show that the algorithm can achieve rapi tracking of the robot system. Keywors: Robot Passivity Dissipativity Robustness. Introuction Robot system is a very complex multi-input multi-output nonlinear system with time-varying an the strong coupling of nonlinear ynamics. issipativity theory has been put forwar by the people's concern an reache a wie range of research results (Van er Schaft A.J. 999; Feng 998) in the 970s. Its substance is that the internal energy system loss is always less than the external energy supply rate. H control (Mei003)an passive control are a special cases of issipative control. he H control is a kin of interference suppression control which can not only guarantee the stability of the system but also achieve smallest egree requeste by the interference to system output. If the supply rate is the prouct of input an output the state will be passivity problem. Passivity theory has been wiely use in many engineering problems such as electrical systems an thermal power systems. A strict passive ynamic system generally has excellent ynamic characteristics an satisfactory robustness. (Feng 999pp ) Stabilization controller is esigne by using passivity theory on the efinite part of the robot system an then uses the issipativity theory for interference suppression uner the conition of existing moel error an external isturbance. he scheme enhances the robustness of the robot tracking at the same time an improves the tracking accuracy an spee.. Dissipative Control for Robot. Moel Control Law Design Consiering the following n-egree of robot the ynamic equation is as follows Mqq () Cqqq () Gq () () 95

2 Vol. No. 4 Moern Applie Science Where the vector qt () is the n joint angle; M( q ) is the n n symmetric positive efinite inertia matrix; Cqqq ( ) is the n vector of coriolis an centrifugal torques; Gq ( ) is the n vector of gravitational torques; is the n vector of actuator joint torques. he escription of robot systems in Eq () has the following characteristics: Property : M( q ) is a boune symmetric positive efinite matrix its inverse matrix is a boune. here are positive number an 0 I M I. Property : M ( q) C( q q ) is skew symmetric matrix both M an C in Eq () satisfy the following equations x M ( q) C( q q ) x 0 M C( q q ) C( q q ) Suppose that the esire trajectory of system is escribe by q q an q then the corresponing error is efine as eq q eqq e q q he can be given as u is a control input signal. um( qq ) Cqqq ( ) Gq ( ) () For system in Eq() we can give non-linear compensation in Eq() receive error ynamic equation: Me Ce u (3) he state vector is efine: (4) Obviously when lim e 0 lim e 0 if an only if t t x ex ee lim xt ( ) 0 (5) t In the new coorinates an Eq(3) can be translate into the following equation of state: x xx Mx ( M C) x( M C) x u he output signal is efine: (6) y x (7) Choose the form of Lyapunov function as follows: V( t x) x x x M( q) x (8) Along the state trajectory an its time erivative is as follows: he feeback control law is as follows With 96 Vtx ( ) x xx Mqx ( ) x Mqx ( ) x x x M( x x ) Cx u x M ( q) Cx x x x M( x x ) Cx x u u ( x) v (9) ( x) M( x x ) Cx x

3 Moern Applie Science July 008 Vtx x xyvyv v So ( ) his shows that the close-loop system is passive from the input viz. v to the output viz. y. Accoring to relations of passivity an asymptotic stability let vy x the close-loop system is graual an stable. he equation of state (6) is replace by: x xx x M ( xcx) M v y x If formula (0) is passive which can be written for the general form: x f( x) g( x) v y h( x) Accoring to a KYP lemma then: (0) () V f ( x ) 0 x () V g ( x ) h ( x ) x. Robust Controller Design Consier the moel error an external isturbances the robot moel: M() q q C() q q q G() q (3) he equation of state: x xx (4) Mx ( M C) x( M C) x u Combining (9) an (4) we have x xx x (5) M ( xcx) M ( v) y x heorem: If the efinite part (0) of system (5) is the passive for any positive number system (5) is the issipative for the supply rate uner the conition of the negative feeback: s( y) y vy y (6) 4 where V( x) is that the energy storage function. Proof: give a general form for system (5) x f( x) g( x)( v) y h( x) For any positive number combining formula (6) hence V LfV LgV( v) LfV y y y y y 4 LV f y y y y y 4 4 LV y y f (7) So the close-loop system is issipative on supply rate s( wy ) y. ( ) V x is the storage function. 97

4 Vol. No. 4 Moern Applie Science he close-loop system is issipative on s( wy ) y supply rate. hat is to say the is rejection ratio of close-loop system from interference to output y that is H control. 3. Simulation Research In orer to verify the control strategy the objects are simulate base on the MIMO ynamics moel of two-dof robot manipulator (Liu 005). he ynamics moel is given by Mqq () Cqqq () Gq () ut () where ml m( l l) mll cos( q) ml mll cos( q) mllq sin( q) mll ( q q )sin( q) Mq () Cqq () ml mll cos( q) ml mllq sin( q) 0 ( mm) lgcos( q) mlgcos( qq) gq ( ) mlg cos( q q) m kg m kg l m l m g 9.8 is acceleration of gravity 4 he moel error an external isturbance: t qq t q q t () sin() cos() he position orers of joint an joint : q sin(t) ; q cos(t) he initial joint angle position: q (0) q (0) he initial angular spee: q (0) q (0) 0 0 he rejection ratio from interference to output y : 0.03 Simulation results show that the application of issipative control algorithm can ensure the smooth controller an graual convergence of the tracking error. 4. Conclusion In this paper consiering this robot system in moel error an external isturbances conitions robust H tracking controller is esigne by using isposition theory. he H tracking controller can effectively inhibit interference an the robot system can achieve quickly an accurately tracking. References Feng Chunbo an Fei Shumin. (004). Robust control of nonlinear systems. Beijing:Science Press. Feng ChunboZhang Kanjian an Fei Shumin. (999). Passivity-base esign of robust control systems. ACA AUOMAICA SINICA 5(5) Liu Jinkun (005). MALAB Simulation for Sliing Moe Control. Beijing: singhua University Press. Mei Shengwei Shen ielong an Liu Kangzhi. (003). Moern Robust Control heory an Application. Beijing: singhua University Press. Van er Schaft A.J. (999). L -gain an Passivity echniques in Nonlinear Control. Lonon: Springer-Ver-lag. 98

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