Abstract. This paper proposes a new method for guaranteed integration of state
equations. Within this framework, the variables of interest are trajectories
submitted to both arithmetic and differential equations. The approach consists
in formalizing a problem thanks to a constraint network and then apply these
constraints to sets of trajectories.
The contribution of the paper is to provide a reliable framework to enclose the
solutions of these differential equations. Its use is shown
to be simple, more general and more competitive than existing approaches dealing
with guaranteed integration, especially when applied to mobile robotics. The
flexibility of the developed framework allows to deal with non-linear differential
equations or even differential inclusions built from datasets, while considering
observations of the states of interest. An illustration of this method
is given over several examples with mobile robots.

Keywords: guaranteed integration, tube programming, mobile robotics, constraints, contractors, ODE

Tube: definition

A tube is defined
as an envelope enclosing an uncertain trajectory $\mathbf{x}(\cdot):\mathbb{R}\rightarrow\mathbb{R}^{n}$.
In this paper, a tube $[\mathbf{x}](\cdot)$ is an interval of two trajectories
$[\mathbf{x}^{-}(\cdot),\mathbf{x}^{+}(\cdot)]$ such that $\forall t,~\mathbf{x}^{-}(t)\leqslant\mathbf{x}^{+}(t)$.
A trajectory $\mathbf{x}(\cdot)$ belongs to the tube $[\mathbf{x}](\cdot)$
if $\forall t,\mathbf{~x}(t)\in[\mathbf{x}](t)$.
Figure 1 illustrates a tube implemented with
a set of boxes.