By An-chyau Huang
This publication introduces an unified functionality approximation method of the regulate of doubtful robotic manipulators containing normal 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. in spite of actuator dynamics, the unified method remains to be possible. these kinds of gains make the ebook stick out from different current guides.
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Additional info for Adaptive Control of Robot Manipulators: A Unified Regressor-free Approach
Then (2) is asymptotically stable if Vɺ does not vanish identically along any trajectory of (2) other than the trivial solution x = 0 . The result is global if the properties hold for the entire state space and V ( x) is radially unbounded. Lyapunov stability theorem for non-autonomous systems Given the non-autonomous system (1) with an equilibrium point at the origin, and let N be a neighborhood of the origin, then the origin is (i) stable if 38 Chapter 2 Preliminaries ∀x ∈ N , there exists a scalar function V (x, t ) such that V (x, t ) > 0 and Vɺ (x, t ) ≤ 0 ; (ii) uniformly stable if V (x, t ) > 0 and decrescent and Vɺ (x, t ) ≤ 0 ; (iii) asymptotically stable if V (x, t ) > 0 and Vɺ (x, t ) < 0 ; (iv) globally n asymptotically stable if ∀x ∈ℜ , there exists a scalar function V (x, t ) such that V (x, t ) > 0 and Vɺ (x, t ) < 0 and V (x, t ) is radially unbounded; (v) uniformly asymptotically stable if ∀x ∈ N , there exists a scalar function V (x, t ) such that V (x, t ) > 0 and decrescent and Vɺ (x, t ) < 0 ; (vi) globally n uniformly asymptotically stable if ∀x ∈ℜ , there exists a scalar function V (x, t ) such that V (x, t ) > 0 and decrescent and is radially unbounded and Vɺ (x, t ) < 0 ; (vii) exponentially stable if there exits α , β , γ > 0 such that 2 2 2 ∀ x ∈ N , α x ≤ V ( x, t ) ≤ β x and Vɺ (x, t ) ≤ −γ x ; (viii) globally exponentially stable if it is exponentially stable and V (x, t ) is radially unbounded.
In this section, we are going to review the sliding controller design including two smoothing techniques to eliminate the chattering activity in the control effort. Let us consider a non-autonomous system x ( n ) = f (x, t ) + g (x, t )u + d (t ) (1) where x = [ x xɺ ⋯ x ( n −1) ] ∈ℜ is the state vector, x∈ℜ the output of interest, and u(t)∈ℜ the control input. The function f (x,t)∈ℜ and the disturbance d(t)∈ℜ are both unknown functions of time, but bounds of their variations should be available.
A matrix A ∈ℜ n × n is nonsingular if ∃B ∈ℜ n × n such that AB = BA = I where I is an n × n identity matrix. If B exists, then it is known as the inverse of A and is denoted by A −1 . The inverse operation has the following properties: ( A −1 ) −1 = A (2a) ( A T ) −1 = ( A −1 )T (2b) (α A) −1 = 1 α A −1 ∀α ∈ℜ, α ≠ 0 ( AB) −1 = B −1A −1 ∀B ∈ℜ n × n with valid inverse (2c) (2d) A matrix A ∈ℜ n × n is said to be positive semi-definite (denoted by A ≥ 0 ) if x Ax ≥ 0 ∀x ∈ℜ n . It is positive definite (denoted by A > 0 ) if x T Ax > 0 ∀x ∈ℜ n , x ≠ 0 .
Adaptive Control of Robot Manipulators: A Unified Regressor-free Approach by An-chyau Huang