Our intention in preparing this book was to present in a possibly simple
manner these branches of the error analysis which find direct applications
in solving various problem in engineering practice. The main reason for
writing this text was the lack of such an approach in existing books dealing
with the error calculus. Most of the books are devoted to mathematical
statistics and to probability theory. The range of applications is usually
limited to the problems of general statistics and to the analysis of errors in
various measuring techniques.
Much less attention is paid in these books to two-dimensional and threedimensional
distributions, and almost no attention is given to the problems
connected with the two-dimensional and three-dimensional vectorial
functions of independent random variables. The theory of such vectorial
functions finds new applications connected, for example, with the analysis
of the positioning accuracy of various mechanisms, among them of robot
manipulators and automatically controlled earth-moving and loading machines,
such as excavators.
Besides the basic information concerning the classical simple applications
of error calculus, a substantial part of the book is devoted to new
aspects of more advanced problems along with numerous examples of practical
applications in engineering practice. Among others, the Mohr circles
representation of tensors is used for transformation of the components of
covariance tensors, for determination of linear regression, for the analysis
of the accuracy of artillery fire, and for the analysis of the positioning accuracy
of various mechanisms. The methods of determination of the ellipses
and ellipsoids of probability concentration have been described in detail,
along with examples of practical calculations.
Chapters I, II, and III contain presentation of the fundamentals of error
calculus: basic characteristics of error distributions, histograms and their
various applications, basic continuous distributions of errors and functions
of independent random variables. In Chapter IV, two-dimensional distributions
of errors are discussed with applications to the analysis of the accuracy
of artillery fire, to the determination of linear regression for the sets of
experimental points, and to the calculation of correlation coefficients. Fundamentals
of the theory of two-dimensional continuous independent and
dependent random variables are also discussed in that chapter. Then the
methods of determination of the ellipses of probability concentration for
the two-dimensional continuous normal distribution are given.
Chapter V deals with the two-dimensional vectorial functions of independent
random variables along with practical applications to the analysis of
the positioning accuracy of mechanisms with two-dimensional movements.
The procedure of determination of ellipses of probability concentration is
also described.
In Chapter VI, the three-dimensional distributions of errors are considered,
while Chapter VII deals with the three-dimensional vectorial functions
of independent random variables. The theory is illustrated by examples of
the analysis of the positioning accuracy of robot manipulators. The examples
of determining the ellipsoids of probability concentration are presented.
This book has been written for readers whose main interests are applications
of error calculus in various problems of engineering.We have indicated
that certain important concepts of that calculus such as, for example, the
variance and covariance, are notionally analogous to the concepts of inertia
moments of plane or solid figures. The standard deviation is analogous to
the so-called inertia radius of such figures. The procedure for calculating
such values is analogous to that of determination of the centers of gravity
of plane or solid figures.
In all seven chapters much attention is paid to the practical significance of
error analysis. However, some additional information concerning its mathematical
foundations has been included in this book. It may be omitted
by readers who are mainly interested in applications of error calculus. For
convenience, the sections containing such material are marked with an asterisk.
Authors want to express their thanks to Professors Marek Sokolowski
and Richard Hetnarski for their help and discussions during preparation of
this book.