The robot presented in the previous sections has numerous advantages
but its performances are very sensitive to its geometry (i.e. to the
location of the points ). The previous sections have
presented analysis problems: analyze for a given robot what are
its performance. But we may also consider an even more complex problem
which is the synthesis problem i.e. find what should be the
geometry of the robot such that it satisfies some performance
criteria.
This is very complex issue but one which has clearly a large impact in
practice (for example when designing a robot whose load will be over 2
tons and whose accuracy should be better than a micrometer as
considered by the European Synchrotron Radiation Facility in
Grenoble).
Consider for example the following problem: what should be the
geometry of the robot (i.e. the location of the and the
limits
on the
) such that a given set
of position/orientation
can be reached by the
platform ?
The inverse kinematic analysis allows to obtain the constraints
as a set of inequalities
between the design
parameters
and the elements of the set
. A classical
approach to solve this problem will be to determine the set
that minimize the cost function
.
But this approach has many drawbacks:
Using this approach we get not only one solution but ranges for the
solution: indeed each point of the solution boxes satisfy all
inequalities which means that the theoretical robot will be able
to reach all the elements of
. But we may take also the
manufacturing errors
into account. Indeed for a range
A=
in a solution
box we may consider the range
AP=
that is included in
the previous range and exists as the ranges of a solution box have a
width at least
. If we choose as design parameter a point
in AP, then for the real robot the design parameter value will be
included in A. As this is true for all the design parameters, then we
may guarantee that the real robot will be able to reach all elements
of
.
Using this approach we are able to find all the geometries such that the real robot satisfy the workspace constraints. Now we may consider another design criteria and use the same approach, that will lead to another set of design parameters. Taking the intersection of this set with the set obtained for the workspace we will be able to compute all robot geometries that satisfy the new design criteria and the workspace constraint.
Here again interval analysis is an elegant approach that allows to solve a very practical problem. This approach has been used by the COPRIN project in various industrial contracts.