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**Some examples of what can be done with interval analysis**

**Date**: Version 0.1, July 20002

**Date**: Version 0.2, November 2002

**Date**: Version 0.3, March 2003

**Date**: Version 0.4, May 2004

**Date**: Version 1.0, September 2006

## 1 Introduction

An introduction (in french) to the methods
used in interval analysis with some classical and non classical examples
## 2 Solving problems

### 2.1 Inequalities system

An introduction
### 2.2 Systems of dimension greater than 0

An example (in french)
## 3 Non solving problems

### 3.1 Roots of parametric polynomials

Introduction (in french)
## 4 Applications

### 4.1 Design of a steering mechanism

An example of systems solving, error analysis and how to deal with
dimension 1 variety (pdf, in English):
the example

### 4.2 Forward kinematics of a parallel robot

An example of system solving for a very difficult robotics problem
(pdf, in English):
the example

### 4.3 Workspace calculation for a parallel robot

A difficult example to show how it is possible to compute an
approximation of a 6 dimensional variety whose unknowns must satisfy a
set of inequalities
postscript
pdf

It may also be of interest to determine the part of the workspace
where the robot is *well conditioned*. A robot has sensors that
measure its internal states and that are used to control the position
of the hand. Errors measurement of the sensors induce error in the
positioning of the hand. We may look for the part of the workspace in
which the amplification factor between the sensors errors and the
positioning errors is included in a given range:
pdf

### 4.4 Trajectory verification for robots

A trajectory for the hand of a robot may be defined as analytical
functions of time. Here we have to check that a given trajectory is
included in the workspace of the robot (which is defined by a set of
inequalities). An interesting point is that interval analysis allows
to take into account errors in the modeling of the robot:
pdf

### 4.5 Parallel robot singularities

There is a linear relation ship between the velocities (translational
and angular) X of the end-effector of a robot and the velocities T ot
its actuator, which may be written as T=JX where J is called
traditionally as the *jacobian matrix* of the robot. This jacobian
is usually known in closed-form for parallel robots.

A problem occur when this matrix become singular. Indeed for a zero
velocity of the actuators, the velocity of the end-effector can be no
more 0: even if the actuators are locked the end-effector is still
moving and the robot control has been lost. Furthermore in that case
the forces/torques to which are submitted the actuators may go to
infinity, leading to a breakdown of the robot. Hence it is essential
to be able to determine if such case may occur within a given
workspace for the end-effector.

This is a complex issue as even if the jacobian is known in
closed-form, it may be impossible to calculate analytically its
determinant due to its very large size or complexity. Furthermore
geometrical parameters of the robot appears in the jacobian, which are
known only with some uncertainties. It
is therefore essential to be able to manage these uncertainties.

Interval analysis allows to solve this problem and to the best of our
knowledge it's the only method that is able to certify the a workspace
is singularity-free whatever are the uncertainties in the robot
model. But to get a really efficient algorithm it is necessary to
master complex tests about the regularity of matrices whose elements
are intervals: pdf

### 4.6 Optimal design of robots

A robot is defined by a set of geometrical parameter (such as lengths
of links, vectors defining axis of rotation,..) and the choice of
these parameters has a very large influence on the performances of the
robot. We look here for values of these parameters so that the
workspace of the robot will include a set of pre-defined position
while at the same time the positioning errors will be lower than a
given threshold. The important point here is that we are able to
calculate an approximation of all the possible parameters values. This
allow to take into account manufacturing errors and thus to guarantee
that the physical robot derived from the theoretical solutions will
satisfy the constraints.:
pdf

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*This document was translated from L*^{A}T_{E}X by
*H*^{E}V^{E}A*.
*