SU\G(𝔸)/K·U
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All published works (39)

Enabling safe walking rehabilitation on the exoskeleton Atalante: experimental results

2023-05-29
Maxime Brunet , Marine Pétriaux , Florent Di Meglio +1 more
This paper exposes a control architecture enabling rehabilitation of walking impaired patients with the lower-limb exoskeleton Atalante. Atalante's control system is modified to allow the patient to contribute to the … This paper exposes a control architecture enabling rehabilitation of walking impaired patients with the lower-limb exoskeleton Atalante. Atalante's control system is modified to allow the patient to contribute to the walking motion through their efforts. Only the swing leg degree of freedom along the nominal path is relaxed. An online trajectory optimization checks that the muscle forces do not jeopardize stability. The optimization generates reference trajectories that satisfy several key constraints from the current point to the end of the step. One of the constraints requires that the center or pressure remains inside the support polygon, which ensures that the support leg subsystem successfully tracks the reference trajectory. As a result of the presented works, the robot provides a non-zero force in the direction of motion only when required, helping the patient go fast enough to maintain balance (or preventing him from going too fast). Experimental results are reported. They illustrate that variations of ±50% of the duration of the step can be achieved in response to the patient's efforts and that many steps are achieved without falling.
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Enabling safe walking rehabilitation on the exoskeleton Atalante: experimental results

2023-01-01
Maxime Brunet , Marine Pétriaux , Florent Di Meglio +1 more
This paper exposes a control architecture enabling rehabilitation of walking impaired patients with the lower-limb exoskeleton Atalante. Atalante's control system is modified to allow the patient to contribute to the … This paper exposes a control architecture enabling rehabilitation of walking impaired patients with the lower-limb exoskeleton Atalante. Atalante's control system is modified to allow the patient to contribute to the walking motion through their efforts. Only the swing leg degree of freedom along the nominal path is relaxed. An online trajectory optimization checks that the muscle forces do not jeopardize stability. The optimization generates reference trajectories that satisfy several key constraints from the current point to the end of the step. One of the constraints requires that the center or pressure remains inside the support polygon, which ensures that the support leg subsystem successfully tracks the reference trajectory. As a result of the presented works, the robot provides a non-zero force in the direction of motion only when required, helping the patient go fast enough to maintain balance (or preventing him from going too fast). Experimental results are reported. They illustrate that variations of $\pm$50% of the duration of the step can be achieved in response to the patient's efforts and that many steps are achieved without falling. A video of the experiments can be viewed at https://youtu.be/_1A-2nLy5ZE
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Fast replanning of a lower-limb exoskeleton trajectories for rehabilitation

2022-12-06
Maxime Brunet , Marine Pétriaux , Florent Di Meglio +1 more
The paper addresses the rehabilitation of disabled patients using a lower-limb fully-actuated exoskeleton. We propose a novel numerical method to replan the current step without jeopardizing stability. Stability is evaluated … The paper addresses the rehabilitation of disabled patients using a lower-limb fully-actuated exoskeleton. We propose a novel numerical method to replan the current step without jeopardizing stability. Stability is evaluated in the light of a simple linear time-invariant surrogate model. The method's core is the analysis of an input-constrained optimal control problem with state specified at an unspecified terminal time. A detailed study of the extremals given by Pontryagin Maximum Principle is sufficient to characterize its feasibility. This allows a fast replanning strategy. The efficiency of the numerical algorithm (resolution time below 1 ms) yields responsiveness to the patient's request. Realistic simulations on a full-body model of the patient-exoskeleton system stress that cases of practical interest for physiotherapists are well-addressed.
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The multi-variable Affine Index Polynomial for tangles

2022-01-01
Nicolas Petit
The multi-variable affine index polynomial was defined by the author in previous work. The aim of this short note is to update the definition so it is generalizable to virtual … The multi-variable affine index polynomial was defined by the author in previous work. The aim of this short note is to update the definition so it is generalizable to virtual tangles and to show it is compatible with tangle decomposition. We also introduce the Turaev moves for virtual tangles, and discuss how to recover the weight of each crossing as an intersection pairing of homology classes.
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Fast replanning of a lower-limb exoskeleton trajectories for rehabilitation

2022-01-01
Maxime Brunet , Marine Pétriaux , Florent Di Meglio +1 more
The paper addresses the rehabilitation of disabled patients using a lower-limb fully-actuated exoskeleton. We propose a novel numerical method to replan the current step without jeopardizing stability. Stability is evaluated … The paper addresses the rehabilitation of disabled patients using a lower-limb fully-actuated exoskeleton. We propose a novel numerical method to replan the current step without jeopardizing stability. Stability is evaluated in the light of a simple linear time-invariant surrogate model. The method's core is the analysis of an input-constrained optimal control problem with state specified at an unspecified terminal time. A detailed study of the extremals given by Pontryagin Maximum Principle is sufficient to characterize its feasibility. This allows a fast replanning strategy. The efficiency of the numerical algorithm (resolution time below 1 ms) yields responsiveness to the patient's request. Realistic simulations on a full-body model of the patient-exoskeleton system stress that cases of practical interest for physiotherapists are well-addressed.
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Maximum likelihood trajectory for the velocity of an artillery shell in free-flight

2021-05-04
R Séailles , Nicolas Petit
The paper considers the problem of determining the translational velocity of an artillery shell from noisy information. To optimize the smoothness of the estimate which is to be used in … The paper considers the problem of determining the translational velocity of an artillery shell from noisy information. To optimize the smoothness of the estimate which is to be used in a attitude observer through an analysis of its slope, the unknown is sought after under the form of an element of a family of possible curves, solely parameterized by their initial condition. The nonlinear dynamics of the velocity being a contraction, it is shown that a measurement bias is observable. The estimator statistical properties are studied when the number of measurement is increased.
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Tracking Controller Design for Satellite Attitude Under Unknown Constant Disturbance Using Stable Embedding

2020-12-09
Wonshick Ko , Karmvir Singh Phogat , Nicolas Petit +1 more
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Optimal Control of Systems Subject to Input-Dependent Hydraulic Delays

2020-03-23
Charles-Henri Clerget , Nicolas Petit
In this article, we study the optimal control of systems subject to input-varying hydraulic delays, i.e., systems where the delay on the input depends on the past values of the … In this article, we study the optimal control of systems subject to input-varying hydraulic delays, i.e., systems where the delay on the input depends on the past values of the input through a specific integral relation. The calculus of variations of this problem reveals its nondifferentiable nature. Then, a smooth relaxation is proposed to derive an iterative optimization algorithm. A convergence proof is detailed. The practical interest of the algorithm is evidenced on a numerical example.

Tracking Controller Design for Satellite Attitude Under Unknown Constant Disturbance Using Stable Embedding

2020-01-01
Wonshick Ko , Karmvir Singh Phogat , Nicolas Petit +1 more
We propose a tracking control law for the fully actuated rigid body system in the presence of any unknown constant disturbance by employing quaternions with the stable embedding technique and … We propose a tracking control law for the fully actuated rigid body system in the presence of any unknown constant disturbance by employing quaternions with the stable embedding technique and Lyapunov stability theory. The stable embedding technique extends the attitude dynamics from the set of unit quaternions to the set of quaternions, which is a Euclidean space, such that the set of unit quaternions is an invariant set of the extended dynamics. Such a stable extension of the system dynamics to a Euclidean space allows us to employ well studied Lyapunov techniques in Euclidean spaces such as LaSalle-Yoshizawa's theorem. A robust tracking control law is proposed for the attitude dynamics subject to unknown constant disturbance and the convergence properties of the tracking control law is rigorously proven. It is demonstrated with the help of numerical simulations that the proposed control law has a remarkable performance even in some challenging situations.
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Dynamic optimization of processes with time varying hydraulic delays

2019-09-17
Charles-Henri Clerget , Nicolas Petit

Slow gas flow passing a solid is a convection/diffusion equation

2019-01-01
Delphine Bresch‐Pietri , Nicolas Petit
This paper exposes a formal result showing that a set of two coupled hyperbolic equations governing the thermal exchanges between a gas passing by a solid is actually close, in … This paper exposes a formal result showing that a set of two coupled hyperbolic equations governing the thermal exchanges between a gas passing by a solid is actually close, in a detailed mathematical sense, to a single convection/diffusion equation. Exhaust gas passing by a solid catalyst is a typical example of such a situation. This result, the key derivation steps of which are given here, bridges the gap between the two formulations, which have received distinct types of contributions by the control community in the recent years.

The Multi-variable Affine Index Polynomial

2019-01-01
Nicolas Petit
We define a multi-variable version of the Affine Index Polynomial for virtual links. This invariant reduces to the original Affine Index Polynomial in the case of virtual knots, and also … We define a multi-variable version of the Affine Index Polynomial for virtual links. This invariant reduces to the original Affine Index Polynomial in the case of virtual knots, and also generalizes the version for compatible virtual links recently developed by L. Kauffman. We prove that this invariant is a Vassiliev invariant of order one, and study what happens as we shift the coloring of one or more components.
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The Wriggle polynomial for virtual tangles

2019-01-01
Nicolas Petit
We generalize the Wriggle polynomial, first introduced by L. Folwaczny and L. Kauffman, to the case of virtual tangles. This generalization naturally arises when considering the self-crossings of the tangle. … We generalize the Wriggle polynomial, first introduced by L. Folwaczny and L. Kauffman, to the case of virtual tangles. This generalization naturally arises when considering the self-crossings of the tangle. We prove that the generalization (and, by corollary, the original polynomial) are Vassiliev invariants of order one for virtual knots, and study some simple properties related to the connected sum of tangles.
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Angular velocity nonlinear observer from vector measurements

2016-11-02
Lionel Magnis , Nicolas Petit
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Implicit Integral Equations for Modeling Systems with a Transport Delay

2016-01-01
Delphine Bresch‐Pietri , Nicolas Petit
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Angular Velocity Nonlinear Observer From Single Vector Measurements

2015-11-17
Lionel Magnis , Nicolas Petit
The paper proposes a technique to estimate the angular velocity of a rigid body from single vector measurements. Compared to the approaches presented in the literature, it does not use … The paper proposes a technique to estimate the angular velocity of a rigid body from single vector measurements. Compared to the approaches presented in the literature, it does not use attitude information nor rate gyros as inputs. Instead, vector measurements are directly filtered through a nonlinear observer estimating the angular velocity. Convergence is established using a detailed analysis of a linear-time varying dynamics appearing in the estimation error equation. This equation stems from the classic Euler equations and measurement equations. As is proven, the case of free-rotation allows one to relax the persistence of excitation assumption. Simulation results are provided to illustrate the method.
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Angular velocity nonlinear observer from single vector measurements

2015-03-10
Lionel Magnis , Nicolas Petit
The paper proposes a technique to estimate the angular velocity of a rigid body from single vector measurements. Compared to the approaches presented in the literature, it does not use … The paper proposes a technique to estimate the angular velocity of a rigid body from single vector measurements. Compared to the approaches presented in the literature, it does not use attitude information nor rate gyros as inputs. Instead, vector measurements are directly filtered through a nonlinear observer estimating the angular velocity. Convergence is established using a detailed analysis of a linear-time varying dynamics appearing in the estimation error equation. This equation stems from the classic Euler equations and measurement equations. As is proven, the case of free-rotation allows one to relax the persistence of excitation assumption. Simulation results are provided to illustrate the method.
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Angular velocity nonlinear observer from vector measurements

2015-03-10
Lionel Magnis , Nicolas Petit
The paper proposes a technique to estimate the angular velocity of a rigid body from vector measurements. Compared to the approaches presented in the literature, it does not use attitude … The paper proposes a technique to estimate the angular velocity of a rigid body from vector measurements. Compared to the approaches presented in the literature, it does not use attitude information nor rate gyros as inputs. Instead, vector measurements are directly filtered through a nonlinear observer estimating the angular velocity. Convergence is established using a detailed analysis of the linear-time varying dynamics appearing in the estimation error equation. This equation stems from the classic Euler equations and measurement equations. A high gain design allows to establish local uniform exponential convergence. Simulation results are provided to illustrate the method.
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Finite-Type Invariants of Order One for Framed Virtual Knots

2015-01-01
Nicolas Petit
A. Henrich proved the existence of the universal finite-type invariant of order one for virtual knots. We extend the construction and the methods of her paper to framed virtual knots. … A. Henrich proved the existence of the universal finite-type invariant of order one for virtual knots. We extend the construction and the methods of her paper to framed virtual knots. To do so, we introduce the notions of virtual strings and based matrices for framed flat virtual knots.
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Angular velocity nonlinear observer from vector measurements

2015-01-01
Lionel Magnis , Nicolas Petit
The paper proposes a technique to estimate the angular velocity of a rigid body from vector measurements. Compared to the approaches presented in the literature, it does not use attitude … The paper proposes a technique to estimate the angular velocity of a rigid body from vector measurements. Compared to the approaches presented in the literature, it does not use attitude information nor rate gyros as inputs. Instead, vector measurements are directly filtered through a nonlinear observer estimating the angular velocity. Convergence is established using a detailed analysis of the linear-time varying dynamics appearing in the estimation error equation. This equation stems from the classic Euler equations and measurement equations. A high gain design allows to establish local uniform exponential convergence. Simulation results are provided to illustrate the method.
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Angular velocity nonlinear observer from single vector measurements

2015-01-01
Lionel Magnis , Nicolas Petit
The paper proposes a technique to estimate the angular velocity of a rigid body from single vector measurements. Compared to the approaches presented in the literature, it does not use … The paper proposes a technique to estimate the angular velocity of a rigid body from single vector measurements. Compared to the approaches presented in the literature, it does not use attitude information nor rate gyros as inputs. Instead, vector measurements are directly filtered through a nonlinear observer estimating the angular velocity. Convergence is established using a detailed analysis of a linear-time varying dynamics appearing in the estimation error equation. This equation stems from the classic Euler equations and measurement equations. As is proven, the case of free-rotation allows one to relax the persistence of excitation assumption. Simulation results are provided to illustrate the method.
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Backstepping stabilization of an underactuated 3 × 3 linear hyperbolic system of fluid flow equations

2012-06-27
Florent Di Meglio , Rafael Vázquez , Miroslav Krstić +1 more
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Backstepping stabilization of an underactuated 3 × 3 linear hyperbolic system of fluid flow equations

2012-06-01
Florent Di Meglio , Rafael Vázquez , Miroslav Krstić +1 more
We investigate the boundary stabilization of a particular subset of 3×3 linear hyperbolic systems with varying coefficients on a bounded domain. The system is underactuated since only one of the … We investigate the boundary stabilization of a particular subset of 3×3 linear hyperbolic systems with varying coefficients on a bounded domain. The system is underactuated since only one of the three hyperbolic PDEs is actuated at the boundary. The setup considered in the paper occurs in control of multiphase flows on oil production systems. We use a backstepping approach to design a full-state feedback law yielding exponential stability of the origin.
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Adaptive backstepping for uncertain systems with time-delay on-line update laws

2011-06-01
Delphine Bresch‐Pietri , Jonathan Chauvin , Nicolas Petit
This paper addresses the general problem of the equilibrium regulation of potentially unstable linear systems with an unknown input time-delay and unknown parameters in the plant. We extend recent results … This paper addresses the general problem of the equilibrium regulation of potentially unstable linear systems with an unknown input time-delay and unknown parameters in the plant. We extend recent results from the literature where such systems are treated using a backstepping approach applied to a distributed parameters system representation of the delay. Furthering previous approaches, our contribution concerns on-line adaptation of the delay. We develop a local asymptotic convergence result for single-input systems. It is illustrated in simulations on the control of the Air-Fuel Ratio in Spark Ignition engines. The results on this particular example stress the merits of the proposed control algorithm which, with reduced implementation difficulties, reveals sympathetic to on-line applications.
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Adaptive backstepping controller for uncertain systems with unknown input time-delay. Application to SI engines

2010-12-01
Delphine Bresch‐Pietri , Jonathan Chauvin , Nicolas Petit
In this paper, we study the equilibrium regulation of potentially unstable linear systems with an unknown input time-delay and unknown parameters in the plant. We extend recent results from the … In this paper, we study the equilibrium regulation of potentially unstable linear systems with an unknown input time-delay and unknown parameters in the plant. We extend recent results from the literature where such systems are treated using a backstepping approach applied to a distributed parameters system representation of the delay. We develop a local result, robust to delay errors and apply it for the control of the Air-Fuel Ratio in Spark Ignition engines. A proof of convergence is established for this particular example. Experimental results stress the relevance of the proposed control algorithm.
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Slugging in multiphase flow as a mixed initial-boundary value problem for a hyperbolic system

2010-11-08
Florent Di Meglio , Glenn Ole Kaasa , Nicolas Petit +1 more
This paper studies the multiphasis slugging flow phenomenon occurring in oil wells and flow lines. The main contribution is a low-dimensional distributed parameters model, comprising as states the gas mass … This paper studies the multiphasis slugging flow phenomenon occurring in oil wells and flow lines. The main contribution is a low-dimensional distributed parameters model, comprising as states the gas mass fraction, the pressure, and gas velocity. Along with appropriate boundary conditions, on the one-dimensional space domain, it constitutes a well-posed mixed initial-boundary value problem for a quasilinear hyperbolic system. Numerical simulation results obtained with a presented characteristics method solver stress the validity of the approach and the fair representativeness of the model. In particular, the period of simulated oscillations and their overall shape is in accordance with reference results from the literature. Controllability and observability open problems are exposed for future works.
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Control Problems for One-Dimensional Fluids and Reactive Fluids with Moving Interfaces

2010-01-01
Nicolas Petit
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Iterative calibration method for inertial and magnetic sensors

2009-12-01
Éric Dorveaux , David Vissière , Alain-Pierre Martin +1 more
We address the problem of three-axis sensor calibration. Our focus is on magnetometers. Usual errors (misalignment, non-orthogonality, scale factors, biases) are accounted for. We consider a method where no specific … We address the problem of three-axis sensor calibration. Our focus is on magnetometers. Usual errors (misalignment, non-orthogonality, scale factors, biases) are accounted for. We consider a method where no specific calibration hardware is required. We solely use the fact that the norm of the sensed field must remain constant irrespective of the sensors orientation. The proposed algorithm is iterative. Its convergence is studied. Experiments conducted with MEMS sensors (magnetometers) stress the relevance of the approach.
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Ascent optimization for a heavy space launcher

2009-08-01
C. Ponssard , Knut Graichen , Nicolas Petit +1 more
This paper exposes work to optimize the ascent trajectory of a multistage space launcher. Motivated by well-known singular arc effects as in the case of the Goddard rocket, we investigate … This paper exposes work to optimize the ascent trajectory of a multistage space launcher. Motivated by well-known singular arc effects as in the case of the Goddard rocket, we investigate whether the payload of the launcher can be enhanced by means of a variable thrust during the first phase of the flight. Due to the high complexity and numerical sensitivity of the multistage problem, we use constructive methods for initialization and handling input constraints in optimal control to compute the ascent trajectories. Based on realistic numerical values for key design parameters and flight dynamics, we conclude that the computed trajectories are in fact full thrust.

Incorporating a class of constraints into the dynamics of optimal control problems

2009-04-22
Knut Graichen , Nicolas Petit
Abstract A method is proposed to systematically transform a constrained optimal control problem (OCP) into an unconstrained OCP, which can be treated in the standard calculus of variations. The considered … Abstract A method is proposed to systematically transform a constrained optimal control problem (OCP) into an unconstrained OCP, which can be treated in the standard calculus of variations. The considered class of constraints comprises up to m input constraints and m state constraints with well‐defined relative degree, where m denotes the number of inputs of the given nonlinear system. Starting from an equivalent normal form representation, the constraints are incorporated into a new system dynamics by means of saturation functions and differentiation along the normal form cascade. This procedure leads to a new unconstrained OCP, where an additional penalty term is introduced to avoid the unboundedness of the saturation function arguments if the original constraints are touched. The penalty parameter has to be successively reduced to converge to the original optimal solution. The approach is independent of the method used to solve the new unconstrained OCP. In particular, the constraints cannot be violated during the numerical solution and a successive reduction of the constraints is possible, e.g. to start from an unconstrained solution. Two examples in the single and multiple input case illustrate the potential of the approach. For these examples, a collocation method is used to solve the boundary value problems stemming from the optimality conditions. Copyright © 2009 John Wiley & Sons, Ltd.
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Constructive Methods for Initialization and Handling Mixed State-Input Constraints in Optimal Control

2008-08-22
Knut Graichen , Nicolas Petit
Newmethodsarepresented toaddresstwoissuesin indirectoptimalcontrol:the calculationofastarting pointfor the numerical solution and the consideration of mixed state-input constraints. In the first method, an auxiliary optimal control problem is constructed from a given initial … Newmethodsarepresented toaddresstwoissuesin indirectoptimalcontrol:the calculationofastarting pointfor the numerical solution and the consideration of mixed state-input constraints. In the first method, an auxiliary optimal control problem is constructed from a given initial trajectory of the system. Its adjoint variables are simply zero. This auxiliary problem is then used within a homotopy approach to eventually reach the original optimal control problem and the desired optimal solution. The second method concerns the incorporation of mixed stateinput constraints into the dynamics of the considered optimal control problem. It uses saturation functions which strictly satisfy the constraints. In this way, the original constrained optimal control problem is transformed into an unconstrained one with an additional regularization term. The two approaches are derived within a general framework. For sake of illustration, they are applied to the space shuttle reentry problem, which represents a challenging benchmark due to its high numerical sensitivity and the presence of input and heating constraints. The reentryproblemissolvedwithacollocationmethodanddemonstratestheapplicabilityandaccuracyoftheproposed constructive methods.
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A continuation approach to state and adjoint calculation in optimal control applied to the reentry problem

2008-01-01
Knut Graichen , Nicolas Petit

CASING-HEADING PHENOMENON IN GAS-LIFTED WELL AS A LIMIT CYCLE OF A 2D MODEL WITH SWITCHES

2005-01-01
Laure Sinègre , Nicolas Petit , P. Lemetayer +2 more

Boundary control of a nonlinear Stefan problem

2004-07-08
William B. Dunbar , Nicolas Petit , Pierre Rouchon +1 more
The classical Stefan problem is a linear one-dimensional heat equation with a free boundary at one end, modelling a column of liquid (e.g. water) in contact with an infinite strip … The classical Stefan problem is a linear one-dimensional heat equation with a free boundary at one end, modelling a column of liquid (e.g. water) in contact with an infinite strip of solid (ice). Given the fixed boundary conditions, the column temperature and free boundary motion can be uniquely determined. In the inverse problem, one specifies the free boundary motion, say from one steady-state length to another, and seeks to determine the column temperature and fixed boundary conditions, or boundary control. This motion planning problem is a simplified version of a crystal growth problem. In this paper, we consider motion planning of the free boundary (Stefan) problem with a quadratic nonlinear reaction term. The treatment here is a first step towards treating higher order nonlinearities as observed in crystal growth furnaces. Convergence of a series solution is proven and a detailed parametric study on the series radius of convergence given. Moreover, we prove that the parametrization can indeed be used for motion planning purposes; computation of the open loop motion planning is straightforward and we give simulation results.
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Motion Planning for a nonlinear Stefan Problem

2003-02-01
William B. Dunbar , Nicolas Petit , Pierre Rouchon +1 more
In this paper we consider a free boundary problem for a nonlinear parabolic partial differential equation. In particular, we are concerned with the inverse problem, which means we know the … In this paper we consider a free boundary problem for a nonlinear parabolic partial differential equation. In particular, we are concerned with the inverse problem, which means we know the behavior of the free boundary a priori and would like a solution, e.g. a convergent series, in order to determine what the trajectories of the system should be for steady-state to steady-state boundary control. In this paper we combine two issues: the free boundary (Stefan) problem with a quadratic nonlinearity. We prove convergence of a series solution and give a detailed parametric study on the series radius of convergence. Moreover, we prove that the parametrization can indeed can be used for motion planning purposes; computation of the open loop motion planning is straightforward. Simulation results are given and we prove some important properties about the solution. Namely, a weak maximum principle is derived for the dynamics, stating that the maximum is on the boundary. Also, we prove asymptotic positiveness of the solution, a physical requirement over the entire domain, as the transient time from one steady-state to another gets large.
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Motion planning for two classes of nonlinear systems with delays depending on the control

2002-11-28
Nicolas Petit , Y. Creff , Pierre Rouchon
Two classes of nonlinear delayed controlled systems are considered: nonlinear hyperbolic equations (such as the Burgers equation without diffusion) and models of mixing processes with nonnegligible pipe holdups. Both can … Two classes of nonlinear delayed controlled systems are considered: nonlinear hyperbolic equations (such as the Burgers equation without diffusion) and models of mixing processes with nonnegligible pipe holdups. Both can be seen as systems with delays depending on the control. As for flat systems, the trajectories of such systems can be explicitly parameterized. This is achieved by enlarging the set of allowed manipulations (classical algebraic computations and time derivations) by using compositions and inversions of functions. This provides an easy motion planning algorithm.

Dynamics and solutions to some control problems for water-tank systems

2002-04-01
Nicolas Petit , Pierre Rouchon
We consider a tank containing a fluid. The tank is subjected to directly controlled translations and rotations. The fluid motion is described by linearized wave equations under shallow water approximations. … We consider a tank containing a fluid. The tank is subjected to directly controlled translations and rotations. The fluid motion is described by linearized wave equations under shallow water approximations. For irrotational flows, a new variational formulation of Saint-Venant equations is proposed. This provides a simple method to establish the equations when the tank is moving. Several control configurations are studied: one and two horizontal dimensions; tank geometries (straight and nonstraight bottom, rectangular and circular shapes), tank motions (horizontal translations with and without rotations). For each configuration, we prove that the linear approximation is steady-state controllable and provide a simple and flatness-based algorithm for computing the steering open-loop control. These algorithms rely on operational calculus. They lead to second order equations in space variables whose fundamental solutions define delay operators corresponding to convolutions with compact support kernels. For each configuration, several controllability open-problems are proposed and motivated.

Flatness of Heavy Chain Systems

2001-01-01
Nicolas Petit , Pierre Rouchon
In this paper the flatness [M. Fliess, J. Lévine, P. Martin, and P. Rouchon, Internat. J. Control, 61 (1995), pp. 1327--1361, M. Fliess, J. Lévine, P. Martin, and P. Rouchon, … In this paper the flatness [M. Fliess, J. Lévine, P. Martin, and P. Rouchon, Internat. J. Control, 61 (1995), pp. 1327--1361, M. Fliess, J. Lévine, P. Martin, and P. Rouchon, IEEE Trans. Automat. Control, 44 (1999), pp. 922--937] of heavy chain systems, i.e., trolleys carrying a fixed length heavy chain that may carry a load, is addressed in the partial derivatives equations framework. We parameterize the system trajectories by the trajectories of its free end and solve the motion planning problem, namely, steering from one state to another state. When considered as a finite set of small pendulums, these systems were shown to be flat [R. M. Murray, in Proceedings of the IFAC World Congress, San Francisco, CA, 1996, pp. 395--400]. Our study is an extension to the infinite dimensional case. Under small angle approximations, these heavy chain systems are described by a one-dimensional (1D) partial differential wave equation. Dealing with this infinite dimensional description, we show how to get the explicit parameterization of the chain trajectory using (distributed and punctual) advances and delays of its free end. This parameterization results from symbolic computations. Replacing the time derivative by the Laplace variable s yields a second order differential equation in the spatial variable where s is a parameter. Its fundamental solution is, for each point considered along the chain, an entire function of s of exponential type. Moreover, for each, we show that, thanks to the Liouville transformation, this solution satisfies, modulo explicitly computable exponentials of s, the assumptions of the Paley--Wiener theorem. This solution is, in fact, the transfer function from the flat output (the position of the free end of the system) to the whole state of the system. Using an inverse Laplace transform, we end up with an explicit motion planning formula involving both distributed and punctual advances and delays operators.
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δ-Freeness of a class of linear systems

1997-07-01
Nicolas Petit , Y. Creff , Pierre Rouchon
Starting from a simple example of linear delayed system (with 2 inputs and 2 outputs) commonly used in process control, we show that, as for flat systems (see [1]). an … Starting from a simple example of linear delayed system (with 2 inputs and 2 outputs) commonly used in process control, we show that, as for flat systems (see [1]). an explicit parametrization of all the trajectories can be found. Once more this leads to an easy motion planning. More generally speaking, we prove that this property, called δ-freeness (see [2. 4]) is general among higher dimensions linear delayed systems. More theoretically speaking, we use the module framework and consider a linear delayed system as a finitely generated module over the ring R[d/dt,δ], where δ is one or a set of delay operators. We show that this system is δ-free. That is we can find a basis of its corresponding module over the localized ring R[d/dt, δ, δ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−1</sup> ]. An applicable way to exhibit such a basis is explicitly described.

Common Coauthors

Coauthor Papers Together
Florent Di Meglio 7
Lionel Magnis 6
Pierre Rouchon 6
Knut Graichen 4
Maxime Brunet 4
Marine Pétriaux 4
Delphine Bresch‐Pietri 4
Y. Creff 2
Miroslav Krstić 2
Charles-Henri Clerget 2
Philippe Martin 2
Wonshick Ko 2
Rafael Vázquez 2
Karmvir Singh Phogat 2
William B. Dunbar 2
Dong Eui Chang 2
Jonathan Chauvin 2
Glenn Ole Kaasa 1
Philippe Gervaud 1
Alain-Pierre Martin 1
Éric Dorveaux 1
P. Lemetayer 1
P. Menegatti 1
David Vissière 1
Laure Sinègre 1
Vidar Alstad 1
C. Ponssard 1
J. Laurent-Varin 1
R Séailles 1

Commonly Cited References

A Lie-Backlund approach to equivalence and flatness of nonlinear systems

1999-05-01
Michel Fliess , Jean Lévine , Philippe Martin +1 more
A new system equivalence relation, using the framework of differential geometry of jets and prolongations of infinite order, is studied. In this setting, two systems are said to be equivalent … A new system equivalence relation, using the framework of differential geometry of jets and prolongations of infinite order, is studied. In this setting, two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives. This is a Lie-Backlund isomorphism. The authors prove that, although the state dimension is not preserved, the number of input channels is kept fixed. They also prove that a Lie-Backlund isomorphism can be realized by an endogenous feedback. The differentially flat nonlinear systems introduced by the authors (1992) via differential algebraic techniques, are generalized and the new notion of orbitally flat systems is defined. They correspond to systems which are equivalent to a trivial one, with time preservation or not. The endogenous linearizing feedback is explicitly computed in the case of the VTOL aircraft to track given reference trajectories with stability.
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Controllability and observability of linear delay systems: an algebraic approach

1998-01-01
Michel Fliesś , Hugues Mounier
Interpretations of most existing controllability and observability notions for linear delay systems are given. Module theoretic characterizations are presented. This setting enables a clear and precise comparison of the various … Interpretations of most existing controllability and observability notions for linear delay systems are given. Module theoretic characterizations are presented. This setting enables a clear and precise comparison of the various examined notions. A new notion of controllability is introduced, which is called pi-freeness.
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One-dimensional automotive catalyst modeling

2005-01-01
Christopher Depcik , Dennis N. Assanis

Motion planning for the heat equation

2000-01-01
Béatrice Laroche , Philippe Martin , Pierre Rouchon
The paper gives an explicit open-loop control, able to approximately steer the one-dimensional heat equation with control on the boundary from any state to any other state. The control is … The paper gives an explicit open-loop control, able to approximately steer the one-dimensional heat equation with control on the boundary from any state to any other state. The control is obtained thanks to a parametrization of the solutions of the heat equation by a series involving infinitely many derivatives of the system 'flat output'. Copyright © 2000 John Wiley & Sons, Ltd.

Kalman filtering of spacecraft attitude and the QUEST model

1990-01-01
M.D. Shuster

Parabolic equations in one space variable and the non‐characteristic cauchy problem

1967-08-01
C. Denson Hill

Real and Complex Analysis.

1987-01-01
G. A. Garreau , Walter Rudin
Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive … Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Linear systems with delayed controls: A reduction

1982-08-01
Zvi Artstein
Linear systems with delayed control action are transformed into systems without delays. Under an absolute continuity condition, the new system is an ordinary differential control equation. In the general case, … Linear systems with delayed control action are transformed into systems without delays. Under an absolute continuity condition, the new system is an ordinary differential control equation. In the general case, the new system is a measure-differential control system. It is shown how the controllability, stabilization, and various optimization problems can be analyzed via the reduced systems.

Minimizing the maximum heating of a re-entering space shuttle: An optimal control problem with multiple control constraints

1996-01-01
H. Kreim , Bernd Kugelmann , Hans Josef Pesch +1 more
This paper presents a numerical investigation of an optimal re-entry manoeuvre under several control and control-state constraints. The essential aim of the optimization is the minimization of the maximal skin … This paper presents a numerical investigation of an optimal re-entry manoeuvre under several control and control-state constraints. The essential aim of the optimization is the minimization of the maximal skin temperature of an orbiter. It is demonstrated that the interaction of different solution techniques is indispensable in order to successfully treat such a highly constrained problem. The reduction of the skin temperature is significant. Moreover, the maximum heat flux and the integrated heat flux are also reduced considerably by the optimization.

Motion planning for two classes of nonlinear systems with delays depending on the control

2002-11-28
Nicolas Petit , Y. Creff , Pierre Rouchon
Two classes of nonlinear delayed controlled systems are considered: nonlinear hyperbolic equations (such as the Burgers equation without diffusion) and models of mixing processes with nonnegligible pipe holdups. Both can … Two classes of nonlinear delayed controlled systems are considered: nonlinear hyperbolic equations (such as the Burgers equation without diffusion) and models of mixing processes with nonnegligible pipe holdups. Both can be seen as systems with delays depending on the control. As for flat systems, the trajectories of such systems can be explicitly parameterized. This is achieved by enlarging the set of allowed manipulations (classical algebraic computations and time derivations) by using compositions and inversions of functions. This provides an easy motion planning algorithm.

Suboptimal control of systems with multiple delays

1980-04-01
M. Malek‐Zavarei

Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays

2008-04-09
Miroslav Krstić , Andrey Smyshlyaev

Sur la nature analytique des solutions des équations aux dérivées partielles. Premier mémoire

1918-01-01
Maurice Gevrey
ou impression de ce fichier doit contenir la présente mention de copyright.Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques ou impression de ce fichier doit contenir la présente mention de copyright.Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
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Optimal control problems with delays in state and control variables subject to mixed control–state constraints

2008-03-19
Laurenz Göllmann , Daniela Kern , H. Maurer
Abstract Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control–state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum … Abstract Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control–state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods are discussed by which the delayed optimal control problem is transformed into a large‐scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and numerical examples from chemical engineering and economics illustrate the results. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Controllability of delay-differential systems

1990-01-01
Paula Rocha , Jan C. Willems

A BVP solver based on residual control and the Maltab PSE

2001-09-01
Jacek A. Kierzenka , L. F. Shampine
Our goal was to make it as easy as possible to solve a large class of boundary value problems (BVPs) for ordinary differential equations in the Matlab problem solving environment … Our goal was to make it as easy as possible to solve a large class of boundary value problems (BVPs) for ordinary differential equations in the Matlab problem solving environment (PSE). We present here theoretical and software developments resulting in bvp4c, a capable BVP solver that is exceptionally easy to use.

General theory of partial differential equations and microlocal analysis : proceedings of the Workshop on General Theory of PDEs and Microlocal Analysis, International Centre for Theoretical Physics, Trieste 1995

1996-01-01
QI Min-you , Luigi Rodino
General theory of ODE and Gevrey classes, Chen Hua and Luigi Rodino pseudo-differential operators - an introduction, Otto Liess coherent states and evolution equations, Nicolas Lerner non-linear microlocal analysis, Xu … General theory of ODE and Gevrey classes, Chen Hua and Luigi Rodino pseudo-differential operators - an introduction, Otto Liess coherent states and evolution equations, Nicolas Lerner non-linear microlocal analysis, Xu Chao-Jing propagation of sungularities and second microlocalization, Chen Shuxing analytic hyperbolic systems, Jean Vaillant.

Stability properties of Smith dead-time compensator controllers

1980-12-01
Z.J. Palmor
Abstract The special stability properties of SISO systems incorporating Smith dead-time compensator controllers are investigated. It is shown that the conventional approach to the design of such systems may lead … Abstract The special stability properties of SISO systems incorporating Smith dead-time compensator controllers are investigated. It is shown that the conventional approach to the design of such systems may lead to practical instabilities. Necessary conditions for practical stability and sufficient conditions guaranteeing certain sensitivity and relative stability properties are stated and proven. The inaccuracies in the models are measured by means of an ' ignorance function '. Sufficient conditions expressed in terms of this function, from which conservative estimates on allowed mismatches in the models may be found, are stated. Illustrative examples demonstrating the main results are presented.

Noether's symmetry Theorem for variational and optimal control problems with time delay

2012-08-01
Gastão S. F. Frederico , Delfim F. M. Torres
We extend the DuBois--Reymond necessary optimality condition andNoether's symmetry theorem to the time delay variational setting.Both Lagrangian and Hamiltonian versions of Noether's theorem areproved, covering problems of the calculus of … We extend the DuBois--Reymond necessary optimality condition andNoether's symmetry theorem to the time delay variational setting.Both Lagrangian and Hamiltonian versions of Noether's theorem areproved, covering problems of the calculus of variationsand optimal control with delays.
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On controller design for systems on manifolds in Euclidean space

2018-08-08
Dong Eui Chang
A new method is developed to design controllers in Euclidean space for systems defined on manifolds. The idea is to embed the state-space manifold $M$ of a given control system … A new method is developed to design controllers in Euclidean space for systems defined on manifolds. The idea is to embed the state-space manifold $M$ of a given control system into some Euclidean space $\mathbb R^n$, extend the system from $M$ to the ambient space $\mathbb R^n$, and modify it outside $M$ to add transversal stability to $M$ in the final dynamics in $\mathbb R^n$. Controllers are designed for the final system in the ambient space $\mathbb R^n$. Then, their restriction to $M$ produces controllers for the original system on $M$. This method has the merit that only one single global Cartesian coordinate system in the ambient space $\mathbb R^n$ is used for controller synthesis, and any controller design method in $\mathbb R^n$, such as the linearization method, can be globally applied for the controller synthesis. The proposed method is successfully applied to the tracking problem for the following two benchmark systems: the fully actuated rigid body system and the quadcopter drone system.
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Constructive Methods for Initialization and Handling Mixed State-Input Constraints in Optimal Control

2008-08-22
Knut Graichen , Nicolas Petit
Newmethodsarepresented toaddresstwoissuesin indirectoptimalcontrol:the calculationofastarting pointfor the numerical solution and the consideration of mixed state-input constraints. In the first method, an auxiliary optimal control problem is constructed from a given initial … Newmethodsarepresented toaddresstwoissuesin indirectoptimalcontrol:the calculationofastarting pointfor the numerical solution and the consideration of mixed state-input constraints. In the first method, an auxiliary optimal control problem is constructed from a given initial trajectory of the system. Its adjoint variables are simply zero. This auxiliary problem is then used within a homotopy approach to eventually reach the original optimal control problem and the desired optimal solution. The second method concerns the incorporation of mixed stateinput constraints into the dynamics of the considered optimal control problem. It uses saturation functions which strictly satisfy the constraints. In this way, the original constrained optimal control problem is transformed into an unconstrained one with an additional regularization term. The two approaches are derived within a general framework. For sake of illustration, they are applied to the space shuttle reentry problem, which represents a challenging benchmark due to its high numerical sensitivity and the presence of input and heating constraints. The reentryproblemissolvedwithacollocationmethodanddemonstratestheapplicabilityandaccuracyoftheproposed constructive methods.
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Angular velocity nonlinear observer from vector measurements

2015-03-10
Lionel Magnis , Nicolas Petit
The paper proposes a technique to estimate the angular velocity of a rigid body from vector measurements. Compared to the approaches presented in the literature, it does not use attitude … The paper proposes a technique to estimate the angular velocity of a rigid body from vector measurements. Compared to the approaches presented in the literature, it does not use attitude information nor rate gyros as inputs. Instead, vector measurements are directly filtered through a nonlinear observer estimating the angular velocity. Convergence is established using a detailed analysis of the linear-time varying dynamics appearing in the estimation error equation. This equation stems from the classic Euler equations and measurement equations. A high gain design allows to establish local uniform exponential convergence. Simulation results are provided to illustrate the method.
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Feedback Control of an Exoskeleton for Paraplegics: Toward Robustly Stable, Hands-Free Dynamic Walking

2018-11-15
Omar Harib , Ayonga Hereid , Ayush Agrawal +7 more
"I will never forget the emotion of my first steps […]," were the words of Fran?oise, the first user during initial trials of the exoskeleton ATALANTE [1]. "I am tall … "I will never forget the emotion of my first steps […]," were the words of Fran?oise, the first user during initial trials of the exoskeleton ATALANTE [1]. "I am tall again!" were the words of Sandy (the fourth user) after standing up in the exoskeleton. During these early tests, complete paraplegic patients dynamically walked up to 10 m without crutches or other assistance using a feedback control method originally invented for bipedal robots. As discussed in "Summary," this article describes the hardware (shown in Figure 1) that was designed to achieve hands-free dynamic walking, the control laws that were deployed (and those being developed) to provide enhanced mobility and robustness, and preliminary test results. In this article, dynamic walking refers to a motion that is orbitally stable as opposed to statically stable.
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Motion Planning for a nonlinear Stefan Problem

2003-02-01
William B. Dunbar , Nicolas Petit , Pierre Rouchon +1 more
In this paper we consider a free boundary problem for a nonlinear parabolic partial differential equation. In particular, we are concerned with the inverse problem, which means we know the … In this paper we consider a free boundary problem for a nonlinear parabolic partial differential equation. In particular, we are concerned with the inverse problem, which means we know the behavior of the free boundary a priori and would like a solution, e.g. a convergent series, in order to determine what the trajectories of the system should be for steady-state to steady-state boundary control. In this paper we combine two issues: the free boundary (Stefan) problem with a quadratic nonlinearity. We prove convergence of a series solution and give a detailed parametric study on the series radius of convergence. Moreover, we prove that the parametrization can indeed can be used for motion planning purposes; computation of the open loop motion planning is straightforward. Simulation results are given and we prove some important properties about the solution. Namely, a weak maximum principle is derived for the dynamics, stating that the maximum is on the boundary. Also, we prove asymptotic positiveness of the solution, a physical requirement over the entire domain, as the transient time from one steady-state to another gets large.
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Stair Climbing Stabilization of the HRP-4 Humanoid Robot using Whole-body Admittance Control

2019-05-01
Stéphane Caron , Abderrahmane Kheddar , Olivier Tempier
We consider dynamic stair climbing with the HRP-4 humanoid robot as part of an Airbus manufacturing use-case demonstrator. We share experimental knowledge gathered so as to achieve this task, which … We consider dynamic stair climbing with the HRP-4 humanoid robot as part of an Airbus manufacturing use-case demonstrator. We share experimental knowledge gathered so as to achieve this task, which HRP-4 had never been challenged to before. In particular, we extend walking stabilization based on linear inverted pendulum tracking [1] by quadratic programming-based wrench distribution and a whole-body admittance controller that applies both end-effector and CoM strategies. While existing stabilizers tend to use either one or the other, our experience suggests that the combination of these two approaches improves tracking performance. We demonstrate this solution in an on-site experiment where HRP4 climbs an industrial staircase with 18.5 cm high steps, and release our walking controller as open source software.
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Optimal Controls for Systems with Time Lag

1968-05-01
A. Halanay
Previous article Next article Optimal Controls for Systems with Time LagA. HalanayA. Halanayhttps://doi.org/10.1137/0306016PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] G. L. Kharatishvili, The maximum principle in the theory of optimal … Previous article Next article Optimal Controls for Systems with Time LagA. HalanayA. Halanayhttps://doi.org/10.1137/0306016PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] G. L. Kharatishvili, The maximum principle in the theory of optimal processes with time lag, Dokl. Akad. Nauk SSSR, 136 (1961), 39–42 Google Scholar[1A] L. S. Pontryagin, , V. G. Boltyanskii, , R. V. Gamkrelidze and , E. F. Mishchenko, The mathematical theory of optimal processes, Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt, Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962viii+360 MR0166037 0102.32001 Google Scholar[2] Avner Friedman, Optimal control for hereditary processes, Arch. Rational Mech. Anal., 15 (1964), 396–416 10.1007/BF00256929 MR0170744 0122.10801 CrossrefISIGoogle Scholar[3] I. A. Oziganova, On the theory of optimal control of systems with time lag, Seminar on Differential Equations with Deviating Arguments, Vol. II, University of the Friendship of Peoples, Moscow, 1963, 136–145, See also: On the theory of optimal control for problems with time lag, thesis, University of the Friendship of Peoples, Moscow, 1966. Google Scholar[4] Magnus R. Hestenes, On variational theory and optimal control theory, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 23–48 MR0184763 0151.12803 LinkGoogle Scholar[5] Rodney D. Driver, Existence and stability of solutions of a delay-differential system, Arch. Rational Mech. Anal., 10 (1962), 401–426 10.1007/BF00281203 MR0141863 0105.30401 CrossrefISIGoogle Scholar[6] M. Namı k Oğuztöreli, Time-lag control systems, Mathematics in Science and Engineering, Vol. 24, Academic Press, New York, 1966xii+323 MR0217394 0143.12101 Google Scholar[7] G. L. Kharatishvili, A. V. Balakrishnan and , L. W. Neustadt, A maximum principle in extremal problems with delaysMathematical Theory of Control (Proc. Conf., Los Angeles, Calif., 1967), Academic Press, New York, 1967, 26–34 MR0256240 0216.17701 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Continuity of Pontryagin Extremals with Respect to Delays in Nonlinear Optimal ControlRiccardo Bonalli, Bruno Hérissé, and Emmanuel Trélat18 April 2019 | SIAM Journal on Control and Optimization, Vol. 57, No. 2AbstractPDF (576 KB)Optimal Control Problems with Time Delays: Constancy of the HamiltonianRichard B. Vinter25 July 2019 | SIAM Journal on Control and Optimization, Vol. 57, No. 4AbstractPDF (525 KB)The Maximum Principle for Optimal Control Problems with Time DelaysA. Boccia and R. B. Vinter19 September 2017 | SIAM Journal on Control and Optimization, Vol. 55, No. 5AbstractPDF (407 KB)The Construction of the Solution of an Optimal Control Problem Described by a Volterra Integral Equation17 February 2012 | SIAM Journal on Control and Optimization, Vol. 21, No. 4AbstractPDF (1468 KB)Optimal Controls with Pseudodelays18 July 2006 | SIAM Journal on Control, Vol. 12, No. 2AbstractPDF (1227 KB)Optimal Control Problems with a System of Integral Equations and Restricted Phase Coordinates18 July 2006 | SIAM Journal on Control, Vol. 10, No. 1AbstractPDF (1661 KB)The Optimization of Trajectories of Linear Functional Differential Equations18 July 2006 | SIAM Journal on Control, Vol. 8, No. 4AbstractPDF (2777 KB) Volume 6, Issue 2| 1968SIAM Journal on Control History Submitted:06 July 1967Published online:18 July 2006 InformationCopyright © 1968 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0306016Article page range:pp. 215-234ISSN (print):0036-1402Publisher:Society for Industrial and Applied Mathematics

Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem

2001-09-30
Марек Фила , Philippe Souplet
We consider a one-phase Stefan problem for the heat equation with a superlinear reaction term. It is known from a previous work (Ghidouche, Souplet, & Tarzia [5]) that all global … We consider a one-phase Stefan problem for the heat equation with a superlinear reaction term. It is known from a previous work (Ghidouche, Souplet, & Tarzia [5]) that all global solutions are bounded and decay uniformly to 0. Moreover, it was shown in Ghidouche, Souplet, & Tarzia [5] that either: (i) the free boundary converges to a finite limit and the solution decays at an exponential rate, or (ii) the free boundary grows up to infinity and the decay rate is at most polynomial, and it was also proved that small data solutions behave like (i).
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An interior penalty method for inequality constrained optimal control problems

1967-08-01
Leon S. Lasdon , Allan D. Waren , R.K. Rice
This paper presents a penalty function approach to the solution of inequality constrained optimal control problems. The method begins with a point interior to the constraint set and approaches the … This paper presents a penalty function approach to the solution of inequality constrained optimal control problems. The method begins with a point interior to the constraint set and approaches the optimum from within, by solving a sequence of problems with only terminal conditions as constraints. Thus, all intermediate solutions satisfy the inequality constraints. Conditions are given which guarantee that the un "constrained" problems have solutions interior to the constraint set and that in the limit these solutions converge to the constrained optimum. For linear systems with convex objective and concave inequalities, the unconstrained problems have the property that any local minimum is global. Further, under these conditions, upper and lower bounds in the optimum are easily available. Three test problems are solved and the results presented.

A Treatise on the Theory of Bessel Functions.

1923-09-01
Matthew Porter , G. N. Watson
1. Bessel functions before 1826 2. The Bessel coefficients 3. Bessel functions 4. Differential equations 5. Miscellaneous properties of Bessel functions 6. Integral representations of Bessel functions 7. Asymptotic expansions … 1. Bessel functions before 1826 2. The Bessel coefficients 3. Bessel functions 4. Differential equations 5. Miscellaneous properties of Bessel functions 6. Integral representations of Bessel functions 7. Asymptotic expansions of Bessel functions 8. Bessel functions of large order 9. Polynomials associated with Bessel functions 10. Functions associated with Bessel functions 11. Addition theorems 12. Definite integrals 13. Infinitive integrals 14. Multiple integrals 15. The zeros of Bessel functions 16. Neumann series and Lommel's functions of two variables 17. Kapteyn series 18. Series of Fourier-Bessel and Dini 19. Schlomlich series 20. The tabulation of Bessel functions Tables of Bessel functions Bibliography Indices.

Numerical Continuation Methods--An Introduction.

1992-07-01
Layne T. Watson , Eugene L. Allgower , Kurt Georg
1 Introduction.- 2 The Basic Principles of Continuation Methods.- 2.1 Implicitly Defined Curves.- 2.2 The Basic Concepts of PC Methods.- 2.3 The Basic Concepts of PL Methods.- 3 Newton's Method … 1 Introduction.- 2 The Basic Principles of Continuation Methods.- 2.1 Implicitly Defined Curves.- 2.2 The Basic Concepts of PC Methods.- 2.3 The Basic Concepts of PL Methods.- 3 Newton's Method as Corrector.- 3.1 Motivation.- 3.2 The Moore-Penrose Inverse in a Special Case.- 3.3 A Newton's Step for Underdetermined Nonlinear Systems.- 3.4 Convergence Properties of Newton's Method.- 4 Solving the Linear Systems.- 4.1 Using a QR Decomposition.- 4.2 Givens Rotations for Obtaining a QR Decomposition.- 4.3 Error Analysis.- 4.4 Scaling of the Dependent Variables.- 4.5 Using LU Decompositions.- 5 Convergence of Euler-Newton-Like Methods.- 5.1 An Approximate Euler-Newton Method.- 5.2 A Convergence Theorem for PC Methods.- 6 Steplength Adaptations for the Predictor.- 6.1 Steplength Adaptation by Asymptotic Expansion.- 6.2 The Steplength Adaptation of Den Heijer & Rheinboldt.- 6.3 Steplength Strategies Involving Variable Order Predictors.- 7 Predictor-Corrector Methods Using Updating.- 7.1 Broyden's Good Update Formula.- 7.2 Broyden Updates Along a Curve.- 8 Detection of Bifurcation Points Along a Curve.- 8.1 Simple Bifurcation Points.- 8.2 Switching Branches Via Perturbation.- 8.3 Branching Off Via the Bifurcation Equation.- 9 Calculating Special Points of the Solution Curve.- 9.1 Introduction.- 9.2 Calculating Zero Points f(c(s)) = 0.- 9.3 Calculating Extremal Points minsf((c(s)).- 10 Large Scale Problems.- 10.1 Introduction.- 10.2 General Large Scale Solvers.- 10.3 Nonlinear Conjugate Gradient Methods as Correctors.- 11 Numerically Implementable Existence Proofs.- 11.1 Preliminary Remarks.- 11.2 An Example of an Implementable Existence Theorem.- 11.3 Several Implementations for Obtaining Brouwer Fixed Points.- 11.4 Global Newton and Global Homotopy Methods.- 11.5 Multiple Solutions.- 11.6 Polynomial Systems.- 11.7 Nonlinear Complementarity.- 11.8 Critical Points and Continuation Methods.- 12 PL Continuation Methods.- 12.1 Introduction.- 12.2 PL Approximations.- 12.3 A PL Algorithm for Tracing H(u) = 0.- 12.4 Numerical Implementation of a PL Continuation Algorithm.- 12.5 Integer Labeling.- 12.6 Truncation Errors.- 13 PL Homotopy Algorithms.- 13.1 Set-Valued Maps.- 13.2 Merrill's Restart Algorithm.- 13.3 Some Triangulations and their Implementations.- 13.4 The Homotopy Algorithm of Eaves & Saigal.- 13.5 Mixing PL and Newton Steps.- 13.6 Automatic Pivots for the Eaves-Saigal Algorithm.- 14 General PL Algorithms on PL Manifolds.- 14.1 PL Manifolds.- 14.2 Orientation and Index.- 14.3 Lemke's Algorithm for the Linear Complementarity Problem.- 14.4 Variable Dimension Algorithms.- 14.5 Exploiting Special Structure.- 15 Approximating Implicitly Defined Manifolds.- 15.1 Introduction.- 15.2 Newton's Method and Orthogonal Decompositions Revisited.- 15.3 The Moving Frame Algorithm.- 15.4 Approximating Manifolds by PL Methods.- 15.5 Approximation Estimates.- 16 Update Methods and their Numerical Stability.- 16.1 Introduction.- 16.2 Updates Using the Sherman-Morrison Formula.- 16.3 QR Factorization.- 16.4 LU Factorization.- P1 A Simple PC Continuation Method.- P2 A PL Homotopy Method.- P3 A Simple Euler-Newton Update Method.- P4 A Continuation Algorithm for Handling Bifurcation.- P5 A PL Surface Generator.- P6 SCOUT - Simplicial Continuation Utilities.- P6.1 Introduction.- P6.2 Computational Algorithms.- P6.3 Interactive Techniques.- P6.4 Commands.- P6.5 Example: Periodic Solutions to a Differential Delay Equation.- Index and Notation.
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Numerical Solution of Optimal Control Problems by Direct Collocation

1993-01-01
Oskar von Stryk
By an appropriate discretization of control and state variables, a constrained optimal control problem is transformed into a finite dimensional nonlinear program which can be solved by standard SQP-methods [10]. … By an appropriate discretization of control and state variables, a constrained optimal control problem is transformed into a finite dimensional nonlinear program which can be solved by standard SQP-methods [10]. Convergence properties of the discretization are derived. Prom a solution of this method known as direct collocation, these properties are used to obtain reliable estimates of adjoint variables. In the presence of active state constraints, these estimates can be significantly improved by including the switching structure of the state constraint into the optimization procedure. Two numerical examples are presented.

Boundary control of a nonlinear Stefan problem

2004-07-08
William B. Dunbar , Nicolas Petit , Pierre Rouchon +1 more
The classical Stefan problem is a linear one-dimensional heat equation with a free boundary at one end, modelling a column of liquid (e.g. water) in contact with an infinite strip … The classical Stefan problem is a linear one-dimensional heat equation with a free boundary at one end, modelling a column of liquid (e.g. water) in contact with an infinite strip of solid (ice). Given the fixed boundary conditions, the column temperature and free boundary motion can be uniquely determined. In the inverse problem, one specifies the free boundary motion, say from one steady-state length to another, and seeks to determine the column temperature and fixed boundary conditions, or boundary control. This motion planning problem is a simplified version of a crystal growth problem. In this paper, we consider motion planning of the free boundary (Stefan) problem with a quadratic nonlinear reaction term. The treatment here is a first step towards treating higher order nonlinearities as observed in crystal growth furnaces. Convergence of a series solution is proven and a detailed parametric study on the series radius of convergence given. Moreover, we prove that the parametrization can indeed be used for motion planning purposes; computation of the open loop motion planning is straightforward and we give simulation results.
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Analysis of descriptor systems using numerical algorithms

1981-02-01
Richard F. Sincovec , A. M. Erisman , E. L. Yip +1 more
In this paper we analyze numerical methods for the solution of the large scale dynamical system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E\dot{y}(t)=Ay(t)+g(t),Y(t_{0})=y_{0}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> are matrices, … In this paper we analyze numerical methods for the solution of the large scale dynamical system <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E\dot{y}(t)=Ay(t)+g(t),Y(t_{0})=y_{0}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> are matrices, possibly singular. Systems of this type have been referred to as implicit systems and more recently as descriptor systems since they arise from formulating system equations in physical variables. Special cases of such systems are algebraic-differential systems. We discuss the numerical advantages of this formulation and identify a class of numerical integration algorithms which have accuracy and stability properties appropriate to descriptor systems and which preserve structure, detect nonsolvable systems, resolve initial value consistency problems, and are applicable to "stiff" descriptor systems. We also present an algorithm for the control of the local truncation error on only the state variables.

Complementary filter design on the special orthogonal group SO(3)

2006-10-04
Robert Mahony , Tarek Hamel , Jean Michel Pflimlin
This paper considers the problem of obtaining high quality attitude extraction and gyros bias estimation from typical low cost intertial measurement units for applications in control of unmanned aerial vehiccles. … This paper considers the problem of obtaining high quality attitude extraction and gyros bias estimation from typical low cost intertial measurement units for applications in control of unmanned aerial vehiccles. Two different non-linear complementary filters are proposed: Direct complementary filter and Passive non-linear complementary filter. Both filters evolve explicity on the special orthogonal group SO(3) and can be expressed in quaternion form for easy implementation. An extension to the passive ocmplementary filter is proposed to provide adaptive gyro bias estimation.
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Prediction-based control of moisture in a convective flow

2015-07-01
Delphine Bresch‐Pietri , Kevin Coulon
This work studies a convective flow system and presents experimental closed-loop results carried out on a test-bench representative of several industrial processes. This test bench consists of a horizontal column … This work studies a convective flow system and presents experimental closed-loop results carried out on a test-bench representative of several industrial processes. This test bench consists of a horizontal column equipped with a mist actuator located at the inlet and fans generating an air flow circulating along the tube. Following our recent theoretical design, we implemented a prediction-based control strategy aiming at stabilizing the mist at the output of the tube actuating on the wind speed. Correspondingly, this set-up involves a transport input-dependent delay (between the inlet and the output of the tube). We propose a control-oriented model, in which the transport delay satisfies an integral equation, and compared our prediction-based design with a conventional Proportional-Integral controller. Experimental results underline the relevance of the proposed approach.
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Angular velocity nonlinear observer from vector measurements

2016-11-02
Lionel Magnis , Nicolas Petit
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Dynamics and solutions to some control problems for water-tank systems

2002-04-01
Nicolas Petit , Pierre Rouchon
We consider a tank containing a fluid. The tank is subjected to directly controlled translations and rotations. The fluid motion is described by linearized wave equations under shallow water approximations. … We consider a tank containing a fluid. The tank is subjected to directly controlled translations and rotations. The fluid motion is described by linearized wave equations under shallow water approximations. For irrotational flows, a new variational formulation of Saint-Venant equations is proposed. This provides a simple method to establish the equations when the tank is moving. Several control configurations are studied: one and two horizontal dimensions; tank geometries (straight and nonstraight bottom, rectangular and circular shapes), tank motions (horizontal translations with and without rotations). For each configuration, we prove that the linear approximation is steady-state controllable and provide a simple and flatness-based algorithm for computing the steering open-loop control. These algorithms rely on operational calculus. They lead to second order equations in space variables whose fundamental solutions define delay operators corresponding to convolutions with compact support kernels. For each configuration, several controllability open-problems are proposed and motivated.

Adaptive backstepping controller for uncertain systems with unknown input time-delay. Application to SI engines

2010-12-01
Delphine Bresch‐Pietri , Jonathan Chauvin , Nicolas Petit
In this paper, we study the equilibrium regulation of potentially unstable linear systems with an unknown input time-delay and unknown parameters in the plant. We extend recent results from the … In this paper, we study the equilibrium regulation of potentially unstable linear systems with an unknown input time-delay and unknown parameters in the plant. We extend recent results from the literature where such systems are treated using a backstepping approach applied to a distributed parameters system representation of the delay. We develop a local result, robust to delay errors and apply it for the control of the Air-Fuel Ratio in Spark Ignition engines. A proof of convergence is established for this particular example. Experimental results stress the relevance of the proposed control algorithm.
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Systèmes linéaires sur les opérateurs de Mikusinski et commande d'une poutre flexible

1997-01-01
Michel Fliesś , Hugues Mounier , Pierre Rouchon +1 more
On propose une théorie algébrique des systèmes linéaires sur les opérateurs de Mikusinski, illustrée par la commande d'une poutre flexible d'Euler-Bernoulli. On propose une théorie algébrique des systèmes linéaires sur les opérateurs de Mikusinski, illustrée par la commande d'une poutre flexible d'Euler-Bernoulli.
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No-gap second-order optimality conditions for optimal control problems with a single state constraint and control

2007-07-17
J. Frédéric Bonnans , Audrey Hermant
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Interior methods for constrained optimization

1992-01-01
Margaret H. Wright
Interior methods for optimization were widely used in the 1960s, primarily in the form of barrier methods. However, they were not seriously applied to linear programming because of the dominance … Interior methods for optimization were widely used in the 1960s, primarily in the form of barrier methods. However, they were not seriously applied to linear programming because of the dominance of the simplex method. Barrier methods fell from favour during the 1970s for a variety of reasons, including their apparent inefficiency compared with the best available alternatives. In 1984, Karmarkar's announcement of a fast polynomial-time interior method for linear programming caused tremendous excitement in the field of optimization. A formal connection can be shown between his method and classical barrier methods, which have consequently undergone a renaissance in interest and popularity. Most papers published since 1984 have concentrated on issues of computational complexity in interior methods for linear programming. During the same period, implementations of interior methods have displayed great efficiency in solving many large linear programs of ever-increasing size. Interior methods have also been applied with notable success to nonlinear and combinatorial problems. This paper presents a self-contained survey of major themes in both classical material and recent developments related to the theory and practice of interior methods.

A numerically stable dual method for solving strictly convex quadratic programs

1983-09-01
Donald Goldfarb , A. Idnani

Numerical Continuation Methods

1990-01-01
Eugene L. Allgower , Kurt Georg

CASING-HEADING PHENOMENON IN GAS-LIFTED WELL AS A LIMIT CYCLE OF A 2D MODEL WITH SWITCHES

2005-01-01
Laure Sinègre , Nicolas Petit , P. Lemetayer +2 more

Encyclopedia of Mathematics and its Applications.

1982-07-01
Y. L. L. , William B. Jones , W. J. Thron

Control Problems for One-Dimensional Fluids and Reactive Fluids with Moving Interfaces

2010-01-01
Nicolas Petit
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Multi-contact walking pattern generation based on model preview control of 3D COM accelerations

2016-11-01
Stéphane Caron , Abderrahmane Kheddar
We present a multi-contact walking pattern generator based on preview-control of the 3D acceleration of the center of mass (COM). A key point in the design of our algorithm is … We present a multi-contact walking pattern generator based on preview-control of the 3D acceleration of the center of mass (COM). A key point in the design of our algorithm is the calculation of contact-stability constraints. Thanks to a mathematical observation on the algebraic nature of the frictional wrench cone, we show that the 3D volume of feasible COM accelerations is always an upward-pointing cone. We reduce its computation to a convex hull of (dual) 2D points, for which optimal C(n log n) algorithms are readily available. This reformulation brings a significant speedup compared to previous methods, which allows us to compute time-varying contact-stability criteria fast enough for the control loop. Next, we propose a conservative trajectory-wide contact-stability criterion, which can be derived from COM-acceleration volumes at marginal cost and directly applied in a model-predictive controller. We finally implement this pipeline and exemplify it with the HRP-4 humanoid model in multi-contact dynamically walking scenarios.
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Implicit Integral Equations for Modeling Systems with a Transport Delay

2016-01-01
Delphine Bresch‐Pietri , Nicolas Petit
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The Maximum Principle for Optimal Control Problems with Time Delays**This work was co-funded by the EPSRC grant Control For Energy and Sustainability, grant reference EP/G066477/1 and by the European Union under the 7th Framework Programme “FP7-PEOPLE-2010-ITN”, grant agreement number 264735-SADCO.

2016-01-01
Andrea Boccia , Richard Vinter
In this paper we present necessary conditions for optimal control problems with time delays. The dynamic constraint is formulated as a control delay differential equation, with time delays occuring in … In this paper we present necessary conditions for optimal control problems with time delays. The dynamic constraint is formulated as a control delay differential equation, with time delays occuring in both state and control variables. We allow the dependence of the data on the state variables to be nonsmooth, and the necessary conditions are expressed in terms of set-valued subgradients, in place of conventional derivatives. In the problems considered, the end-time is included in the decision variables. The fact that the end-time is a choice variable is accommodated by a new kind of transversality condition. While nonsmooth necessary conditions have earlier been derived for optimal control problems with time delays, for the most part this has been in the framework of controlled differential inclusions with time delays. By contrast, the necessary conditions of this paper cover general problems involving general, nonsmooth, controlled delay differential equations.

Dynamic walking over rough terrains by nonlinear predictive control of the floating-base inverted pendulum

2017-09-01
Stéphane Caron , Abderrahmane Kheddar
We present a real-time pattern generator for dynamic walking over rough terrains. Our method automatically finds step durations, a critical issue over rough terrains where they depend on terrain topology. … We present a real-time pattern generator for dynamic walking over rough terrains. Our method automatically finds step durations, a critical issue over rough terrains where they depend on terrain topology. To achieve this level of generality, we consider a Floating-base Inverted Pendulum (FIP) model where the center of mass can translate freely and the zero-tilting moment point is allowed to leave the contact surface. This model is equivalent to a linear inverted pendulum with variable center-of-mass height, but its equations of motion remain linear. Our solution then follows three steps: (i) we characterize the FIP contact-stability condition; (ii) we compute feedforward controls by solving a nonlinear optimization over receding-horizon FIP trajectories. Despite running at 30 Hz in a model-predictive fashion, simulations show that the latter is too slow to stabilize dynamic motions. To remedy this, we (iii) linearize FIP feedback control into a constrained linear-quadratic regulator that runs at 300 Hz. We finally demonstrate our solution in simulations with a model of the HRP-4 humanoid robot, including noise and delays over state estimation and foot force control.
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