By An-chyau Huang
This ebook introduces an unified functionality approximation method of the regulate of doubtful robotic manipulators containing common uncertainties. it really works at no cost area monitoring regulate in addition to compliant movement regulate. it really is appropriate to the inflexible robotic and the versatile joint robotic. inspite of actuator dynamics, the unified procedure remains to be possible. these kinds of gains make the e-book stick out from different current courses.
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Extra resources for Adaptive Control of Robot Manipulators: A Unified Regressor-free Approach
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 ) .
Representation 3: In the above representations, all matrix elements are approximated by the same number of orthonormal functions. In many applications, however, it may be desirable to use different number of orthonormal functions for different matrix elements. ,m as f i (x) = w Tf i z f i (10) where w fi , z fi ∈ℜ pi ×1 and pi is the number of terms of the basis functions selected to approximate fi. ,pmax, and then (8) can be expressed in the form p max f ( x) = ∑Wz (14) i i i =1 For approximating the matrix M (t ) ∈ℜ p × q , we may rewrite it into a row p vector as M = [m1 ⋯ m q ] where m i ∈ℜ .
One approach is to use the saturation function sat(σ ) defined below instead of the signum function sgn( s ) . σ if σ ≤ φ sat(σ ) = sgn(σ ) if σ > φ (19) where φ > 0 is called the boundary layer of the sliding surface. , s > φ , the sliding controller with sgn(s) is s exactly the same as the one with sat( ) . Hence, the boundary layer is also φ attractive. When s is inside the boundary layer, equation (10) becomes sɺ + η1 s φ = ∆f + ∆d (20) This implies that the signal s is the output of a stable first-order filter whose input is the bounded model error ∆f + ∆d .
Adaptive Control of Robot Manipulators: A Unified Regressor-free Approach by An-chyau Huang