Nacim Ramdani

Set membership state and parameter estimation for systems described by nonlinear differential equations

This paper investigates the use of guaranteed methods to perform state and parameter estimation for nonlinear continuous-time systems, in a bounded-error context. A state estimator based on a prediction-correction approach is given, where the prediction step consists in a validated integration of an initial value problem for an ordinary differential equation (IVP for ODE) using interval analysis and high-order Taylor models, while the correction step uses a set inversion technique. The state estimator is extended to solve the parameter estimation problem. An illustrative example is presented for each part.

A Hybrid Bounding Method for Computing an Over-Approximation for the Reachable Set of Uncertain Nonlinear Systems

In 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.

Reachability of Uncertain Nonlinear Systems Using a Nonlinear Hybridization

In this paper, we investigate nonlinear reachability computation in presence of model uncertainty, via guaranteed set integration. We show how this can be done by using the classical Mullers existence theorem. The core idea developed is to no longer deal with whole sets but to derive instead two nonlinear dynamical systems which involve no model uncertainty and which bracket in a guaranteed way the space reachable by the original uncertain system. We give a rule for building the bracketing systems. In the general case, the bracketing systems obtained are only piecewise Ck-continuously differential nonlinear systems and hence can naturally be modeled with hybrid automata. We show how to derive the hybrid model and how to address mode switching. An example is given with a biological process.

Complex Interval Arithmetic Using Polar Form

In this paper, the polar representation of complex numbers is extended to complex polar intervals or sectors; detailed algorithms are derived for performing basic arithmetic operations on sectors. While multiplication and division are exactly defined, addition and subtraction are not, and we seek to minimize the pessimism introduced by these operations. Addition is studied as an optimization problem which is analytically solved. The complex interval arithmetic thus defined is illustrated with some numerical examples which show that in many applications, the polar representation is more advisable.

Improving the SAT modulo ODE approach to hybrid systems analysis by combining different enclosure methods

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 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.

Improving SAT modulo ODE for hybrid systems analysis by combining different enclosure methods

Aiming at automatic verification and analysis techniques for hybrid 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 enclosure method for ODEs, and with bracketing systems which exploit monotonicity properties to find enclosures for problems that VNODE-LP alone cannot enclose tightly. We apply our method to the analysis of a non-linear hybrid system by solving predicative encodings of an inductive stability argument and evaluate the impact of different methods and their combination.

Robust dynamic experimental identification of robots with set membership uncertainty

This paper focuses on robust dynamic identification of robots with set membership uncertainty. The error, taken additive on model output (input motor torque), is only assumed to be bounded. Three bounded-error recursive algorithms are implemented to outer-bound the solution parameter set with ellipsoids or parallelotopes. Experimental results are presented for a two degrees-of-freedom SCARA robot.

A structurally optimal control model for predicting and analyzing human postural coordination

This paper proposes a closed-loop optimal control model predicting changes between in-phase and anti-phase postural coordination during standing and related supra-postural activities. The model allows the evaluation of the influence of body dynamics and balance constraints onto the adoption of postural coordination. This model minimizes the instantaneous norm of the joint torques with a controller in the head space, in contrast with classical linear optimal models used in the postural literature and defined in joint space. The balance constraint is addressed with an adaptive ankle torque saturation. Numerical simulations showed that the model was able to predict changes between in-phase and anti-phase postural coordination modes and other non-linear transient dynamics phenomena.

Estimation of the centre of mass from motion capture and force plate recordings: A study on the elderly

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 subjects 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.

Planning and Fast Replanning Safe Motions for Humanoid Robots

This paper introduces effective numerical methods for the planning and fast replanning of safe motions to ensure the safety, balance, and integrity of humanoid robots over the whole motion duration. Our safe methods do not depend on, nor are connected to, any type of modeling or constraints. To plan safe motions, certain constraints have to be satisfied over a continuous interval of time. Classical methods revert to time-grid discretization, which can be risky for the robot. We introduce a hybrid method to plan safe motions, which combines a classical unsafe method with a verification step that checks constraint violation and computes excess by the usage of interval analysis. When the robot meets unexpected situations, it has to replan a new motion, which is often too time consuming. Hence, we introduce a new method to rapidly replan safe motions, i.e., in less than 2 s CPU time. It computes offline feasible subsets in the vicinity of safe motions and finds online a solution in these subsets without actually recomputing the nonlinear constraints. Our methods are validated by the use the HOAP-3 robot, where the motions are run with no balance controller.