ADAPTIVE TRAJECTORY TRACKING CONTROL FOR EULER-LAGRANGE SYSTEMS WITH APPLICATION TO ROBOT MANIPULATORS

Z. Wang,∗ P. Goldsmith,∗∗ and J. Gu∗

References

  1. [1] R. Ortega & M. Spong, Adaptive motion control of rigid robots:A tutorial, Automatica, 25, 1989, 877–888.
  2. [2] R. Ortega, A. Loria, P.J. Nicklasson, & H. Sira-Ramirez,Passivity-based control of Euler-Lagrange Systems (London:Springer-Verlag, 1998).
  3. [3] R. Ortega, A.J. van der Schaft, I. Mareels, & B. Maschke,Putting energy back in control, IEEE Control Systems Maga-zine, 21, 2001, 18–23.
  4. [4] L.U. Gokdere & C.W. Brice, Energy-shaping and input–outputlinearization controllers for induction motors, Proc. of IEEEConf. on Control Applications, Hartford, CT, 1997, 924–926.
  5. [5] R. Ortega & I. Mareels, Energy-balancing passivity-basedcontrol, Proc. American Control Conf., Illinois, USA, 2000,1265–1270.
  6. [6] Z. Wang & P. Goldsmith, Modified energy-balancing controlfor the tracking problem, IET Control Theory Applications, 2,2008, 310–322.
  7. [7] J. Graig, Adaptive control of mechanical manipulators (Read-ing, MA: Addison-Wesley, 1988).
  8. [8] M. Spong & R. Ortega, On adaptive inverse dynamics controlof rigid robots, IEEE Transactions on Automatic Control, 35,1990, 92–95.
  9. [9] R. Middleto & G. Goodwin, Adaptive computed torque controlfor rigid link manipulators, Systems & Control Letters, 10,1988, 9–16.
  10. [10] J. Slotine & W. Li, On the adaptive control of robot manip-ulators, International Journal of Robotics Research, 6, 1987,49–59.
  11. [11] D.E. Koditscheck, Natural motion for robot arms, Proc. 1984IEEE Conf. on Decision and Control, Las Vegas, NV, USA,1984, 733–735.
  12. [12] B. Paden & R. Panja, Globally asymptotically stable pd+controller for robot manipulators, International Journal ofControl, 76, 1988, 1679–1712.
  13. [13] V. Santbanez & R. Kelly, Pd control with feedforward compen-sation for robot manipulators: Analysis and experimentation,Robotica, 19, 2001, 11–19.
  14. [14] R. Kelly, V. Santbanez, & A. Loria, Control of robot manipu-lators in joint space (London: Spinger, 2005).
  15. [15] M. Spong, On the robust control of robot manipulators, IEEETransactions on Automatic Control, 37, 1992, 1782–1786.
  16. [16] G. Tao, On robust adaptive control of robot manipulators,Automatica, 28, 1992, 803–807.
  17. [17] P. Tomei, Robust adaptive control of robots with arbitrarytransient performance and disturbance attenuation, IEEETransactions on Automatic Control, 44, 1999, 654–658.
  18. [18] A. Ibeas & M. Delasen, A robustly stable multiestimation-basedadaptive control scheme for robotic manipulators, Journalof Dynamic Systems, Measurement, and Control, 128, 2006,414–421.
  19. [19] M. Kosaka & H. Shibata, Auto-tuning of adaptive controlwith dead zone, Control and Intelligent Systems, 34 (1), 2006,30–36.
  20. [20] P.R. Ouyang, W.J. Zhang, & M.M. Gupta, Pd-type on-line learning control for systems with state uncertainties andmeasurement disturbances, Control and Intelligent Systems,35 (3), 2007, 1747–1759.54
  21. [21] N. Golea, A. Golea, K. Barra, & T. Bouktir, Observer-based adaptive control of robot manipulators: Fuzzy systemsapproach, Applied Soft Computing, 8, 2008, 778–787.
  22. [22] A.J. van der Schaft, L2-Gain and passivity techniques in non-linear control, Second Edition (London: Springer Communi-cations and Control Engineering series, 2000).A Proof of Eq. (56)As the kinetic energy of EL system (12) has the form of (3),the system (12) can be rewritten in the equivalent form:D(q)¨q + C(q, ˙q) ˙q + g(q) +∂Fp( ˙q)∂ ˙q= u (A.1)whereg(q) =∂Vp∂q(A.2)˙D(q) − 2C(q, ˙q) = 0 (A.3)From (54), we get:˙Tp = ˙eTD(q)¨e +12˙eT ˙D(q)˙e= ˙eT(−C(q, ˙q)˙e − g(q) − Kd ˙e) +12˙eT ˙D(q)˙e− ˙eT(D(q)¨qd + C(q, ˙q) ˙qd)+ ˙eT−∂Y (q)∂qˆ¯θ −∂Y (qd)∂qd˜¯θ − Kp(q − qd)−˙eT∂(Y (q) − Y (qd))Γ1(Y (q) − Y (qd))T2∂e+˙eTddt∂Y1(q, ˙qd)∂ ˙qd+∂Y1(q, ˙qd)∂eˆ¯ϑ+ ˙eT ddt∂(Y1(q, ˙q) − Y1(q, ˙e))∂ ˙qd−˙eT ∂(Y1(q, ˙q) − Y1(q, ˙e))∂e˜ϑ− ˙eTddt∂XT3 Γ2X32∂ ˙e−∂XT3 Γ2X32∂e(A.4)From (A.3) and ddt∂(Y1(q,q) − Y1(q,e))∂e,we get:˙Tp = −˙eTKd ˙e + ˙eT∂(Y (q) − Y (qd))∂e˜¯θ − ˙eTKpe+ ˙eT∂Y1(q, ˙qd)∂e−∂Y1(q, ˙q)∂e+∂Y1(q, ˙e)∂e˜¯ϑ− ˙eT∂(Y (q) − Y (qd))Γ1(Y (q) − Y (qd))T2∂e−˙eTddt∂XT3 Γ2X32∂ ˙e−∂XT3 Γ2X32∂e(A.5)From (52) and (54), we get:˙V = ˙eTKpe +˙˜θTΓ−11˜θ − (Y (q) − Y (qd))˙˜θ− ˙eT ∂(Y (q) − Y (qd))∂e˜θ+ ˙eT∂(Y (q) − Y (qd))Γ1(Y (q) − Y (qd))T2∂e=−˙˜θΓ1˙˜θ + ˙eTKpe − ˙eT ∂(Y (q) − Y (qd))∂e˜θ+ ˙eT∂(Y (q) − Y (qd))Γ1(Y (q) − Y (qd))T2∂e(A.6)From (53) and (55), we get:d(T − Tp)dt= −˙˜ϑTΓ2˙˜ϑ − ˙eT∂XT3∂e˜¯ϑ+ ˙eT ddt∂XT3 Γ2X32∂ ˙e−˙eT ∂XT3 Γ2X32∂e(A.7)From H = T + V and calculating the summation of (A.5),(A.6) and (A.7), we get (56).

Important Links:

Go Back