Abstract
A class of robust tracking controllers for a 6 DOF parallel robot in the presence of nonlinearites and uncertainties are proposed. The controls are based on Lyapunov approach and guarantees practical stability. The controls utilize the information of link displacements and its velocities rather than using the positions or angles of the 6 DOF platform. This can be done by constructing the linkspace coordinates and the workspace coordinates simultaneously by imposing geometric constraints. The controls utilize the possible bound of uncertainty, and the uniform ultimate ball size can be adjusted by a suitable choice of control parameters. The control performance of the proposed algorithms is verified through experiments.
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Abbreviations
- A:
-
Hurwitz matrix
- \(\bar B\), ΔB:
-
Nominal and uncertain input matrix
- B i ,i=1, 2, …, 6:
-
Base joint vector
- d z ,d zi :
-
Uniform bound ball in control system and modified system
- D :
-
Translational vector
- e :
-
Tracking error
- h(·),E(·):
-
Matching function in state and input
- J :
-
Jacobian matrix
- K=[K p K v ],K p1 ,K v1 :
-
Control gain in original system and modified system
- l i :
-
i-th link length
- M(·),C(·),G(·):
-
Inertia, Coriolis, gravitational matrix or vector
- M1(·),C1(·),G1(·):
-
Modified inertia, Coriolis, gravitational matrix or vector
- \(\bar M_1 \), ΔM1:
-
Nominal and uncertain inertia matrix
- p i :
-
Control term compensating uncer-
- P :
-
Positive definite matrix
- P i p ,i=1,2, …, 6:
-
Platform joint vector
- q=[u v w α β γ]T:
-
6 DOF displacement
- Q :
-
Positive semidefinite matrix
- R αβψ :
-
Rotational matrix
- Rz,Rzi:
-
Bounds in control system and modified system
- Si :
-
Weighting in control
- T z ,T zi :
-
Reaching time to uniform ultimate bound in control system and modified system
- u :
-
Control input
- \(\dot y^d \), ÿ d :
-
Desired link velocity and acceleration
- V :
-
Lyapunov function
- z :
-
State variable
- δ e :
-
Uniform stability bound
- λ E (q):
-
Input uncertainty bound
- \(\bar \varepsilon \) :
-
Control gain
- η, η1 :
-
Coefficients in the 2nd order terms
- μ(·):
-
Function utilized in control design
- π(·), π1(·):
-
Bounding functions in original and modified system
- \(\bar \sigma \cdot \underline \sigma \) :
-
Upper and lower bound of inertia matrix
- τ1, τ2 :
-
Minimum and maximum of\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\Omega } _{1i} ,\bar \Omega _{1i} \)
- ϕ(·), ϕ1(·):
-
Uncertain functions in original and modified system
- ζ:
-
State variable
- \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\Omega } _{1i} ,\bar \Omega _{1i} \) :
-
Lower and upper matrices in Lyapunov function
- p :
-
Platform
- T :
-
Transpose of matrix
- d :
-
Desired value
- −1:
-
Inverse matrix
- i :
-
Link index
- p, v :
-
Position and velocity gain
- z :
-
State variable
- 1, 2, 3:
-
Class function on Lyapunov function
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Kim, D.H., Kang, JY. & Lee, KI. Nonlinear robust control design for a 6 DOF parallel robot. KSME International Journal 13, 557–568 (1999). https://doi.org/10.1007/BF03186446
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DOI: https://doi.org/10.1007/BF03186446