We are interested in real-valued system solving (
f(
X) = 0,
f(
X)
0), in optimization problems, and in the proof of the existence of properties
(for example it exists
Xsuch that
f(
X) = 0or it exists two values
X1,
X2such that
f(
X1)>0and
f(
X2)<0). There are few restrictions on the
fwe can deal with as we are able to manage explicit functions using classical mathematical operators (e.g.
sin(
x+
y) + log(cos(
e
x) +
y
2)or implicit functions (e.g. determining if there are parameter values of a parametrized matrix such that the determinant of the matrix is negative, without calculating the
analytical form of the determinant).
Solutions will be searched within a finite domain (called a box) which may be either continuous or mixed (i.e. for which some variables must belong to a continuous range while other variables may only have values within a discrete set). An important point is that we aim to find all the solutions within the domain as soon as the computer arithmetic will allow it: in other words we are looking for certifiedsolutions. For example, for 0-dimensional system solving, we will provide a box that includes one, and only one, solution together with a numerical approximation of this solution, that may further be refined at will using multi-precision.
The kernel of our methods is the use of
interval analysisthat allows one to manipulate expression whose unknowns have interval values. A basic component of interval analysis is the
interval evaluationof an expression. Given an analytical expression
Fin the unknowns
{
x
1,
x
2, ...,
x
n}and ranges
{
X
1,
X
2, ...,
X
n}for these unknowns we are able to compute a range
[
A,
B], called the interval evaluation, such that
{
x1,
x2, ...,
xn}
{
X1,
X2, ...,
Xn}
AF(
x1,
x2, ...,
xn)
B
In other words the interval evaluation provide a lower bound for the minimum of
Fand an upper bound of its maximum over the box.
For example if
F=
xsin(
x+
x2)and
x[0.5, 1.6], then
F([0.5, 1.6]) = [-1.362037441, 1.6], meaning that for any
xin [0.5,0.6] we guarantee that
-1.362037441
f(
x)
1.6.
The interval evaluation of an expression has interesting properties:
it can be implemented in such way that the results are guaranteed with respect to round-off errors i.e. in spite of numerical errors induced by the use of floating point numbers property is still valid
if
A>0or
B<0, then there are now values of the unknowns in their respective ranges that may cancel
F
if
A>0(
B<0), then
Fis positive (negative) for any value of the unknowns in their range
But there is a major drawback of the interval evaluation: there may be an overestimation of
A(
B)i.e. there may be no value of
x1,
x2, ...,
xnsuch that
F(
x1,
x2, ...,
xn) =
A(
B). This overestimation occurs because in our calculation each occurrence of a variable is considered as an independent variable and consequently if a variable have
multiple occurrences, then an overestimation may occur. Such phenomena can be observed in the previous example where
B= 1.6while the real maximum of
Fis approximately 0.9144. The value of
Bis obtained because we are using in our calculation the formula
F=
xsin(
y+
z2)with
y,
zhaving the same interval value than
x.
Fortunately there are methods that allow one to reduce the overestimation and this amount decreases with the width of the ranges. The latter remark leads to the use of a branch-and-bound
strategy in which for a given box a variable range will be bisected, thereby creating two new boxes that will be stored in a list and processed later on. The algorithm will be completed if
all boxes in the list have been processed or if during the process a box generates an answer to the problem at hand (e.g. if we want to prove that
F(
X)<0, then the algorithm stops as soon it is shown that for a box
we have
).
A generic interval analysis algorithm involves the following steps in sequence on the current box:
exclusion operators: these operators determine that there is no solution to the problem within a given box
filters: these operators may reduce the size of the box i.e. decrease the width of the allowed ranges for the variables ,
existence operators: they allow one to determine that there is a unique solution within a given box and are usually associated to a numerical scheme that enable to compute this solution in a safe way
bisection: choose one of the variable and bisect its range for creating two new boxes
storage: store the new boxes in the list
The scope of the COPRIN project is to address all these steps in order to find the most efficient procedures. Our efforts focuses on mathematical developments (adapting classical theorems to interval analysis, proving interval analysis theorems), on the use of symbolic computation and formal proof (a symbolic pre-processing allows one to automatically adapt the solver to the structure of the problem), on software implementation and on experimental tests (for validation purposes).
COPRIN has a long-standing tradition of robotics studies, especially for closed-loop robots , . We address first theoretical issues with the purpose of obtaining analytical and theoretical solutions, but in many cases only numerical solutions can be considered because of the complexity of the problem. This approach has motivated the use of interval analysis for two reasons:
the versatility of interval analysis allows us to address issues that cannot be tackled by any other method (e.g. singularity analysis)
we want to take uncertainties (which are inherent to a robotic device) into account so that we can guarantee that the performance level of the realrobot will satisfy the same properties as the theoreticalone, even in the worst case. This is a crucial issue for many applications in robotics (e.g. medical robot)
Our field of study in robotics focuses on kinematicissues such as workspace and singularity analysis, positioning accuracy, trajectory planning, reliability , , , , and prominently appropriate design, i.e. finding the dimensioning of a robot mechanical architecture that guarantees that the real robot will satisfy a given list of requirements . But the methods that we have developed can be used for other robotic problems, see for example the management of uncertainties in a localization problem with ultrasound .
Our theoretical work must be validated through experiments that are essential for the sake of our credibility. A contrario, experiments will feed the theoretical work (quite often COPRIN has been the first robotic group to address some theoretical issues that were pointed out by experiments). Hence COPRIN works with partners for the development of real robots but also develops its own prototypes (approximately one every 6 years).
In term of applications we have focused on the development of special machines (machine-tool, ultra-high accuracy positioning device, spatial telescope). Although this activity will be pursued we intend to move toward service roboticsi.e. robots that are closer to human activity. In service robotics we are interested in domotics, rehabilitation and medical robots and entertainment that be regrouped under the name of assistive robotics. Compared to special machines for pricing is not an issue (up to a certain point), cost is an important element for service robotics. While we plan to develop simple robotic systems, our work will focus on a different issue: the management of the robot modularity. The mechanical modularity of a robot is obtained by allowing one to change the arrangement of the robot's elements (whose cost may be quite low) so that it is most appropriate for the task. Many such mechanically modular robots are available (or can be designed at will) but finding the right arrangement of the hardware to fulfill the task requirements in spite of mechanical and control uncertainties is an open problem with no known algorithmic solution and developing such algorithms is our long term goal.
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