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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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