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##

Robot workspace analysis

For the robot presented in the previous examples we have a set of
parameters that allow to represent the position/orientation of the
platform. At the same time we have various geometrical parameters that describe
the possible geometry of the robot. Examples of the these parameters
are the distance between the , the angle
between the vector and a fixed direction in space (this
angle represent the amplitude of the motion of the joint we have at
).
In practice all the geometrical parameters are submitted to
constraints. For example the distances must satisfy
, the angle must be such that
where is a unit vector that describes the main orientation
of the ball-and-socket joint at and is the maximal
possible rotation of this joint. All these constraints imply that
the value of the elements of are restricted to lie in a
specific region embedded in the 6-dimensional space, this region being
called the *reachable workspace* of the robot as the points of
this region define all the position/orientation that can be reached by
the platform.

In some cases it is possible to find an analytical description of the
workspace (for example if the orientation is fixed and only the
distance constraints are taken into account) but as soon as all the
constraints are taken into account such analytical form is difficult
to obtain. Interval analysis allows to obtain an approximation of the
workspace as a set of 6-dimensional boxes. It is only an approximation
as the algorithm, based on a classical bisection method, discard boxes
whose width is lower than a given threshold.
At the same time it is possible to obtain a bound on the error between
the volume of the approximation and the volume of the workspace.
For more details see [11].
A variant of the algorithm allows to solve efficiently a very
practical problem: is a given region included in the reachable
workspace of the robot ?

** Next:** Robot synthesis example
** Up:** Examples of applications of
** Previous:** Robot singularity analysis
** Contents**
Jean-Pierre Merlet
2012-12-20