Adaptive fuzzy observer and robust controller for a 2-DOF robot arm

Transcription

1 Adaptive fuzzy observer and robust controller for a 2-DOF robot arm S. Bindiganavile Nagesh Zs. Lendek A. A. Khalate R. Babuška Delft University of Technology, Technical University of Cluj-Napoca FUZZ-IEEE-2012

2 Motivation TS fuzzy models rule base with linear or affine consequents well-established methods, conditions based on LMI unmodelled dynamics adaptive observers Robot manipulators great importance: manufacturing, handling, house-hold tasks human-friendly, compliant mechanical design new control solutions This talk Adaptive observer and robust controller for a 2-DOF robot arm

3 Outline 1 2-DOF Robot arm 2 Adaptive observer 3 Robust controller 4 Estimation in closed-loop 5 Conclusions

4 1 2-DOF Robot arm 2 Adaptive observer 3 Robust controller 4 Estimation in closed-loop 5 Conclusions

5 The arm M R (θ) θ + C R θ = τ M R (θ) = C R (θ, θ) =» P1 + P 2 + 2P 3 cos θ 2 P 2 + P 3 cos θ 2 P 2 + P 3 cos θ 2 P 2» b1 P 3 θ 2 sin θ 2 P 3 ( θ 1 + θ 2 ) sin θ 2 P 3 θ 2 sin θ 2 b 2

6 State-space [ M 1 ẋ = R C ] R 0 x + I 0 x = [ θ 1 θ2 θ 1 θ 2 ] T [ ] M 1 R τ 0 ẋ = A(θ, θ)x + B(θ)τ A 11 A B 11 B 12 A = A 21 A B = B 21 B nonlinearities

7 State-space simplification [ ] b1 0 C R = 0 b 2 ẋ = A(z 1, z 2 )x + B(z 1, z 2 )τ P 2 b 1 z 1 P 2 b 2 z 1 P 3 b 2 z A = P 2 b 1 z 1 P 3 b 1 z 2 b 2 z 1 (P 1 + P 2 ) + 2P 3 b 2 z P 2 k m1 z 1 k m2 (P 2 z 1 + P 3 z 2 ) B = k m1 (P 2 z 1 + P 3 z 2 ) k m2 (2P 3 z 2 + z 1 (P 1 + P 2 )) nonlinearities

8 Comparison

9 TS model Sector nonlinearity z 1 = 1 P 2 3 cos(θ 2) 2 P 1 P 2 z 2 = w 11 (z 1 ) = z 1max z 1 z 1max z 1min w 12 (z 1 ) = 1 w 11 (z 1 ) ẋ = r h i (z)(a i x + B i u) i=1 y = Cx cos(θ 2 ) P 3 2 cos(θ 2 ) 2 P 1 P 2 Uncertainties in the state matrices

10 1 2-DOF Robot arm 2 Adaptive observer 3 Robust controller 4 Estimation in closed-loop 5 Conclusions

11 Adaptive observer To estimate unmodelled dynamics Model: r ẋ = h i (z)(a i x + B i u + M i A δi x) i=1 y = Cx Observer: r x = h i (z)(a i x + B i u + L i (y ŷ) + M i (Âδi x)) i=1 ŷ = C x Adaptive law: Â δi = h i (z)m T i PC e y x T i = 1, 2,..., r

12 Design TS model of the robot arm: 4 rules ẋ = r h i (z)(a i x + B i u + M i A δi x) i=1 y = Cx Uncertainty in the damping coefficients Uncertainty distribution matrix: 1 0 M i = Uncertainty norm: A δi µ max = 2

13 Results

14 Excitation

15 1 2-DOF Robot arm 2 Adaptive observer 3 Robust controller 4 Estimation in closed-loop 5 Conclusions

16 Controller design Model: Control law: ẋ = r h i (z)(a i x + B i u + M i A δi x) i=1 r u = h i (z)f i x i=1 Classical design conditions

17 1 2-DOF Robot arm 2 Adaptive observer 3 Robust controller 4 Estimation in closed-loop 5 Conclusions

18 Results

19 Excitation

20 1 2-DOF Robot arm 2 Adaptive observer 3 Robust controller 4 Estimation in closed-loop 5 Conclusions

21 Conclusions adaptive observer for a 2-DOF robot arm estimation of the uncertainties in closed-loop exploiting the structure of the uncertainty convergence depending on the rule excitation

22 Appendix Acknowledgements This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project number PN-II-RU-TE , contract number 74/