Aiming at automatic verification and analysis techniques for hybrid discrete-continuous systems, we present a novel combination of enclosure methods for ordinary differential equations (ODEs) with the iSAT solver for large Boolean combinations of arithmetic constraints. Improving on our previous work, the contribution of this paper lies in combining iSAT with VNODE-LP, as a state-of-the-art interval solver for ODEs, and with bracketing systems, which exploit monotonicity properties allowing to find enclosures for problems that VNODE-LP alone cannot enclose tightly. We apply the combined iSAT-ODE solver to the analysis of a variety of non-linear hybrid systems by solving predicative encodings of reachability properties and of an inductive stability argument, and evaluate the impact of the different enclosure methods, decision heuristics and their combination. Our experiments include classic benchmarks A preliminary version of this paper appeared in [6]. from the literature, as well as a newly-designed conveyor belt system that combines hybrid behavior of parallel components, a slip-stick friction model with non-linear dynamics and flow invariants and several dimensions of parameterization. In the paper, we also present and evaluate an extension of VNODE-LP tailored to its use as a deduction mechanism within iSAT-ODE, to allow fast re-evaluations of enclosures over arbitrary subranges of the analyzed time span.
International audienceIn this paper, we show how to compute an over-approximation for the reachable set of uncertain nonlinear continuous dynamical systems by using guaranteed set integration. We introduce two ways to do so. The first one is a full interval method which handles whole domains for set computation and relies on state-of-the-art validated numerical integration methods. The second one relies on comparison theorems for differential inequalities in order to bracket the uncertain dynamics between two dynamical systems where there is no uncertainty. Since the derived bracketing systems are piecewise Ck-differentiable functions, validated numerical integration methods cannot be used directly. Hence, our contribution resides in the use of hybrid automata to model the bounding systems. We give a rule for building these automata and we show how to run them and address mode switching in a guaranteed way in order to compute the over approximation for the reachable set. The computational cost of our method is also analyzed and shown to be smaller that the one of classical interval techniques. Sufficient conditions are given which ensure the epsiv-practical stability of the enclosures given by our hybrid bounding method. Two examples are also given which show that the performance of our method is very promising
This paper investigated the possibility of estimating 3D lower limb joint kinematics during five popular rehabilitation exercises of the hip and knee joints based on the data collected from a single inertial measurement unit located on the shank. The leg was modeled as a four-degree-of-freedom serial chain, and the relevant joint angles were represented by Fourier series. A least square approach based on the minimization of the difference between the measured and estimated 3D linear accelerations and angular velocities was used to solve the related analytical problem. The approach was validated on ten healthy young volunteers (ten trials each), comparing the proposed approach with the measurements collected through a stereophotogrammetric system. The average root mean square differences between the estimated joint angles and those reconstructed with the stereophotogrammetric system were inferior than 3.2°with correlation coefficients higher than 0.85.
Index Terms-Inertial measurement unit, motion analysis, hip and knee rehabilitation.Vincent Bonnet received the B.
Summary
To design robust interval observers for uncertain continuous‐time linear systems, a new set‐integration approach is proposed to compute trajectory tubes for the estimation error. Because this approach, the order‐preserving condition on the dynamics of the estimation error is no longer required. Therefore,
scriptH∞ synthesis methods can be used to compute observer gains that reduce the impact of the system uncertainties on the accuracy of the estimated state enclosures. The performance of the proposed approach is showcased through illustrative numerical examples.
The estimation of the centre of mass position in humans is usually based on biomechanical models developed from anthropometric tables. This method can potentially introduce errors in studies involving elderly people, since the ageing process is typically associated with a modification of the distribution of the body mass. In this paper, an alternative technique is proposed, and evaluated with an experimental study on 9 elderly volunteers. The technique is based on a virtual chain, identified from experimental data and locating the subject's centre of mass. Its configuration defines the location of the centre of mass, and is a function of the anatomical joint angles measured on the subject. This method is a valuable investigation tool in the field of geronto-technology, since it overcomes some of the problems encountered with other CoM estimation methods.
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