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Advances in Robot Manipulators Part 2

Tham khảo tài liệu 'advances in robot manipulators part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 32 Advances in Robot Manipulators u 6k Ord h p 16 where h is an estimate of h . Substituting 16 into 8 we may have the system dynamics x p A m x p B m t d T d B p h - h 17 Together with 9 we may have the error dynamics e m A m e m B p h - h 18 er C m e m 19 If we may design an appropriate update law such that h h then 18 implies e m 0 as t n . This further implies T T d as t n . Since D C g and h are functions of time traditional adaptive controllers are not directly applicable. To design the update laws let us apply the function approximation representation15 21 D WD Zd sD C wp ZC sC g wTZg Sg h wT Zh Eh where Wd G np xn Wc G 2pCxn Wg G nPg xn and Wh G WPh xn are weighting matrices ZD G ffl1 Pdxn ZC G n Pcxn Zg G Pe and Zh G nPhx1 are matrices of basis functions and s . are approximation error matrices. The number jB . represents the number of basis functions used. Using the same set of basis functions the corresponding estimates can also be represented as D Wdt Zd g WT Zg C WT zc h wT Zh 20b Define W . W . W . then equation 15 and 18 becomes Ds Cs K d s T T d WD ZDv VT ZC v WT Z s1 21 D g g 33 A Regressor-free Adaptive Control for Flexible-joint Robots based on Function Approximation Technique é A e -B WTZ B E m mm p hh p 2 22 where E1 E1 E D E C E g s q d and E 2 E 2 E h e m are lumped approximation errors. Since Wo are constant vectors their update laws can be easily found by proper selection of the Lyapunov-like function. Let us consider a candidate 1 V s e m Wd Wc Wg Wh 1 sT Ds eTm P e m 1 23 I - Tr WT Q W WT o.w. WT Q w WT Q W 2 1 DqD D CQC C g Qg g h Qh h where P PT G 2 x2 is a positive definite matrix satisfying the Lyapunov equation AmP PAm -CTmCm . The matrices Qd G npx 2pD Qc G n Pcx 2pc Qg G Pg Pg and Qh G i V P nPh are positive definite. The notation Tr . denotes the trace operation of matrices. The time derivative of V along the trajectory of 21 and 22 can be computed as V sT Ds 1 sT Ds eT Pe eT Pem 2 m t m m t m - Tr WTQ W WTOAV WTQW WTOAV r DqD D .