By An-chyau Huang
This publication introduces an unified functionality approximation method of the keep watch over of doubtful robotic manipulators containing basic uncertainties. it really works at no cost house monitoring keep watch over in addition to compliant movement regulate. it's appropriate to the inflexible robotic and the versatile joint robotic. in spite of actuator dynamics, the unified technique remains to be possible. these types of beneficial properties make the publication stand proud of different present courses.
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Additional resources for Adaptive Control of Robot Manipulators: A Unified Regressor-free Approach
Once on the surface, the system behaves like a stable linear system ( d + λ ) n −1 e = 0 ; dt therefore, asymptotic convergence of the tracking error can be obtained. Now, the problem is how to drive the system trajectory to the sliding surface. With s(x, t ) = 0 as the boundary, the state space can be decomposed into two parts: the one with s > 0 and the other with s < 0 . Intuitively, to make the sliding surface attractive, we can design a control u so that s will decrease in the s > 0 region, and it will increase in the s < 0 region.
A simple and powerful tool called Barbalat’s lemma can be used to partially remedy this situation. Let f (t ) be a differentiable function, then Barbalat’s lemma states that if lim f (t ) = k < ∞ and fɺ (t ) is uniformly t→∞ continuous, then lim fɺ (t ) = 0 . It can be proved that a differentiable function is t→∞ uniformly continuous if its derivative is bounded. Hence, the lemma can be rewritten as: if lim f (t ) = k < ∞ and ɺɺ f (t ) exists and is bounded, then t→∞ fɺ → 0 as t → ∞ . In the Lyapunov stability analysis, Barbalat’s lemma can be applied in the fashion similar to La Salle’s theorem: If V (x, t ) is lower bounded, Vɺ ≤ 0 , and Vɺɺ is bounded, then Vɺ → 0 as t → ∞ .
Boundedness of xɺ can be obtained by observing (30a). Therefore, by Barbalat’s lemma, we have proved x → 0 as t → ∞ . n 52 Chapter 2 Preliminaries To prove asymptotic convergence of z, we need to prove that ∀ε > 0 , ∃Tε > 0 such that z (t ) < ε , ∀t ≥ Tε . 6-3), inequality (33) becomes λmax (P ) x(Tε ) + z (Tε ) ≥ z (t ) 2 z (t ) ≤ λmax (P ) x(Tε ) + z (Tε ) 2 2 2 This further implies 2 (34) Since we have proved x → 0 as t → ∞ , this implies that ∀ε > 0 , ∃tε > 0 such that ∀t ≥ tε , x(t ) ≤ ε 2λmax (P ) .