By An-chyau Huang
This booklet introduces an unified functionality approximation method of the keep watch over of doubtful robotic manipulators containing normal uncertainties. it really works at no cost area monitoring regulate in addition to compliant movement regulate. it truly is appropriate to the inflexible robotic and the versatile joint robotic. regardless of actuator dynamics, the unified process continues to be possible. these kinds of positive aspects make the ebook stick out from different latest guides.
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Extra resources for Adaptive Control of Robot Manipulators: A Unified Regressor-free Approach
Since this is not a conventional operation of matrices, dimensions of all involved matrices do T not follow the rule for matrix multiplication. Here, W is a p × kq matrix and Z is a kp × q matrix, but the dimension of M after the operation is still p × q . This notation can be used to facilitate the derivation of update laws. Representation 2: Let us assume that all matrix elements are approximated using the same number, say β, of orthonormal functions, and then the matrix M (t ) ∈ℜ p × q can be represented in the conventional form for matrix multiplications M = WT Z where M , Z ∈ℜ T w 11 0 T 0 w 21 WT = ⋮ ⋮ 0 0 T z11 0 ZT = ⋮ 0 pq β × p ⋯ are in the form T | w 12 0 ⋯ ⋱ 0 ⋮ ⋯ w Tp1 | | | z T21 ⋯ z Tp1 | 0 ⋯ ⋮ 0 (7) 0 ⋮ 0 w T22 ⋮ ⋯ ⋱ 0 ⋮ | ⋯ | | ⋯ | 0 ⋮ w T2 q | ⋯ | 0 0 0 ⋯ 0 ⋱ ⋮ ⋯ w Tpq ⋯ ⋯ 0 | ⋯ | w 1Tq 0 ⋮ 0 0 ⋯ w Tp 2 0 0 ⋯ 0 | ⋯ | 0 0 T z 12 z T22 ⋯ z Tp 2 | ⋯ | 0 0 ⋮ 0 | ⋱ ⋮ | ⋮ ⋮ ⋱ ⋮ | ⋯ | ⋯ 0 | 0 0 ⋯ 0 | ⋯ | z 1Tq ⋮ z T2 q ⋯ 0 ⋯ 0 ⋱ ⋮ ⋯ z Tpq The matrix elements wij and zij are β × 1 vectors.
Two modifications to the update law are introduced to robustify the adaptive loop when the system contains unmodeled dynamics or external disturbances. 1 MRAC of LTI scalar systems Consider a linear time-invariant system described by the differential equation xɺ p = a p x p + b p u (1) where x p ∈ℜ is the state of the plant and u ∈ℜ the control input. The parameters ap and bp are unknown constants, but sgn(bp) is available. The pair (ap, bp) is controllable. The problem is to design a control u and an update law so that all signals in the closed loop plant are bounded and the system output xp tracks the output xm of the reference model xɺ m = a m xm + bm r (2) asymptotically, where am and bm are known constants with a m < 0, and r is a bounded reference signal.
This selection is very easy in implementation, because the robust term is linear in the signal s. For example, controller (9) can be smoothed in the form u= 1 s [−c1eɺ − ⋯ − cn −1e ( n −1) − f m − d m + xd( n ) − η1 ] g φ (21) To justify its effectiveness, the following analysis is performed. , when outside the boundary layer, we may have the result ssɺ ≤ −η s2 φ . Hence, the boundary layer is still attractive. When s is inside the boundary layer, equation (20) can be obtained; therefore, effective chattering elimination can be achieved.