A boundary approach for set inversion
Introduction
Notation. In this paper, a vector of and a vector-valued function will be written in bold font. An interval or an axis aligned box will be written between brackets. The image by of a box will be written as .
Given a function and a set . Set inversion is the problem of characterizing the set of all inside a prior domain such that . We have Note that the function is not necessarily invertible. For instance if , and then Moreover, the solution set maybe empty. For instance, if instead of then The set membership formalism is thus well suited to ill-posed problems involving uncertainties.
It is used in many engineering domains such as localization (Colle and Galerne, 2013, Drevelle and Bonnifait, 2013), parameter estimation (Jaulin et al., 1997, Kreinovich et al., 1997), control (Vinas et al., 2006), calibration (Daney et al., 2006), robotics (Rohou et al., 2019), etc. Most guaranteed algorithms for set inversion are based on interval constraint propagation methods (van Emden, 1999, van Hentenryck et al., 1997). They use a forward–backward contractor (Araya et al., 2008, Chabert and Jaulin, 2009) such as the HC4-revise algorithm (Benhamou et al., 1999a). Now, contractors have mainly be developed for solving equations (Cébério and Granvilliers, 2001, Neumaier, 2004) or global optimization (Hansen, 1992) where the solution set is reduced to some isolate points or surfaces and has no interior.
For set inversion, the solution set has an interior and exterior. Existing contractors are not able to retro-propagate properly the information coming from . This leads to a poor outer approximation and unnecessary computations. It is sometimes possible to use the complementary set to find an inner approximation (Benhamou et al., 1999c), but, a part of the work done for contracting with respect to has to be repeated for , which should be avoided.
In this paper, we propose to inverse the boundary of instead of inverting . Due to the fact that , the corresponding contractor will more efficient. This inversion will provide an outer approximation for the boundary of the solution set . For the parts of the search space which are not in the boundary approximation, we need to show either they are inside or outside . We show that this information can be obtained with almost no extra computation cost by retro-propagating some binary information from . To our knowledge, this is new in the community of constraint propagation.
The paper is organized as follows. Section 2 recalls the principle of set inversion algorithms and motivates the need of a more powerful propagation and the use of the boundaries. Section 3 presents in a new way, inspired from control theory, to present the forward–backward propagation. Section 4 gives the boundary approach for set inversion which corresponds to the main contribution of this paper. Section 5 proposes an application to robot localization based on a Time Difference Of Arrival (TDOA). Section 6 concludes the paper.
Section snippets
Motivation
Fig. 1 illustrates the behavior of a typical contractor-based algorithm (Jaulin and Desrochers, 2014) to characterize a set (magenta). The algorithm uses two types of contractors: a contractor for and a contractor for its complementary . To characterize the part of which is inside a box the principle of the algorithm is the following
- 1.
We have a set of boxes (green in the picture) which have to be studied. These green boxes are qualified as the new boxes. They have been
Principle
We recall here a result given in Jaulin et al. (2001) which can be see as an abstraction of the algorithm presented in Montanari and Rossi (1991). Consider the constraint where is a composition of functions: . The forward–backward sequence given by Algorithm 1 computes exactly the set .
For each , we have where . The proof of this result is a consequence of the fact that the chain structure of
Principle
The boundary of a subset of is the set of points which can be approached both from the inside of and from the outside of . It is also the set of points in the closure of not belonging to its interior. We have the following proposition.
Proposition 4 Consider a continuous function defined everywhere. If , we have
Proof We will show that if then . Denote by the open ball with radius and center . We have
Test-cases
In this Section, we give two examples. The first example illustrates an academic set inversion problem involving a nonlinear function. The Python code is also given. The second example is a localization problem using a TDOA technique (Time Difference Of Arrival). The problem is known to be ill-conditioned and highly nonlinear (Lee, 1975).
Conclusion
In this paper, we have proposed a new forward–backward contraction procedure for set inversion. The principle is to inverse the boundary of the set to be inverted and to retro-propagate a color on the interval bounds in order to decide if the win boxes are inside or outside the solution set.
This approach has several interesting features:
- 1
Contrarily to existing contractor-based set inversion algorithm, we do not have to retro-propagate twice (once for inside , once for outside), but only once
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References (27)
- et al.
Contractor programming
Artificial Intelligence
(2009) - et al.
Mobile robot localization by multiangulation using set inversion
Robot. Auton. Syst.
(2013) - et al.
Introduction to the algebra of separators with application to path planning
Eng. Appl. Artif. Intell.
(2014) - et al.
Constraint relaxation may be perfect
Artificial Intelligence
(1991) - et al.
Exploiting common subexpressions in numerical csps
- et al.
Exploiting monotonicity in interval constraint propagation
- Benhamou, F., Goualard, F., Granvilliers, L., Puget, J.F., 1999a. Revising Hull and Box Consistency, In: Proceedings of...
- et al.
Revising hull and box consistency
- et al.
An algorithm to compute inner approximations of relations for interval constraints
- et al.
Solving nonlinear systems by constraint inversion and interval arithmetic
Interval method for calibration of parallel robots : Vision-based experiments
Mech. Machine Theory Elsevier
Localization confidence domains via set inversion on short-term trajectory
IEEE Trans. Robot.
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