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 house monitoring keep an eye on in addition to compliant movement regulate. it truly is acceptable to the inflexible robotic and the versatile joint robotic. in spite of actuator dynamics, the unified process continues to be possible. these kind of positive aspects make the booklet stick out from different current courses.

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**Extra resources for Adaptive Control of Robot Manipulators: A Unified Regressor-free Approach **

**Sample text**

Examples of orthonormal functions Since there are many areas of applications of orthonormal functions, a sizable body of literature can be easily found. In this section, we consider some of the orthonormal functions that are frequently encountered in engineering problems and useful in our applications. 4 Orthogonal Functions 19 1. Taylor polynomials In the calculus courses, it is well known that given a function f ( x) and a point c in the domain of f, suppose the function is n-times differentiable at c, then we can construct a polynomial Pn ( x) = f (c) + f '(c)( x − c) + +⋯ + f ′′ (c) ( x − c) 2 2!

34 Chapter 2 Preliminaries Since this representation is compatible to all conventional matrix operations, it is used in this book for representing functions, vectors and matrices. 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.

The following are the first 7 polynomials L0 ( x) = 1 L1 ( x) = − x + 1 L2 ( x ) = x 2 − 4 x + 2 (10) 22 Chapter 2 Preliminaries L3 ( x) = − x 3 + 9 x 2 − 18 x + 6 L4 ( x) = x 4 − 16 x 3 + 72 x 2 − 96 x + 24 L5 ( x) = − x 5 + 25 x 4 − 200 x 3 + 600 x 2 − 600 x + 120 L6 ( x) = x 6 − 36 x 5 + 450 x 4 − 2400 x 3 + 5400 x 2 − 4320 x + 720 L7 ( x) = − x 7 + 49 x 6 − 882 x 5 + 7350 x 4 − 29400 x 3 +52920 x 2 − 35280 x + 5040 6. Bessel polynomials The set of Bessel polynomials is orthogonal with respect to the weight function p( x) = x on the interval [0,b] in the form ∫ b 0 xJ n (k i x) J n ( k j x)dx = 0 (11) for all i ≠ j .