# ManiLPV - Identification and robust control of robotic manipulators based on LPV models

## Contents

# Identification and control of flexible manipulators based on LPV models

Methodologies for identification and of milti-input and multi-output linear systems are relatively mature. When the device has a nonlinear nature with smooth variations, one way to handle the variations of the behavior with respect to the operating point is to consider a linear parameter-varying (LPV) model. This project aims at developing efficient methods for identification of LPV models and the synthesis of LPV controllers for flexible manipulators.

*flexible manipulator, robust control, identification, LPV model*

# Staf

## Contacts

## Students

- Daniel Vizer, PhD (defense in 2015)
- Houssem Halalchi, PhD (defense in 2012)

## Partners

Guillaume Mercès (LIAS), Olivier Prot (XLim), Marco Lovera (Politecnico di Milano)

# LPV identification

Rather than developing a LPV models from a physical nonlinear model from linearization or with some change of variables, it might be more convenient to identify the LPV model directly from experimental data. Generally, two kinds of approaches are considered: either the direct identification from measurement and parameters varying over the whole range, either to identify a set of linear time-invariant (LTI) models and interpolate them.

Both ways have been investigated and evaluated in simulation on a 2-segments flexible manipulator considering as available measurements: the joint positions and the position of the end-effector given by a camera.

## Indirect method

In the indirect method, the key issue is the model interpolation. Indeed, the continuity in the variations of the entries of the state matrices requires that the states are properly defined. One solution is to consider a structure derived from the law of Physics and estimate the parameters with an output-error method. However, these methods rely on non-convex optimization and require a good initialization in order to guaranty the quality of the result. In this project, non-smooth optimization algorithms are used in order to obtain an initial structured model from a non structured model.

For the considered flexible manipulator, we have shown that the use of both measurements (joint and end-effector velocities) are necessary in order to obtain a good result.

## A *glocal* method for the direct approach

In the direct approach, it is necessary that the measurements span the range of variation of the scheduling parameter. However, this might be difficult to realize experimentally. For example, in the case where a camera is used in order to measure the speed of the end-effector, a good accuracy requires that an objective with a large focal distance in order to zoom in on the small displacement due to the flexible modes. In the propose *glocal* approach, a set of measurements are done around several operating sets but parameter estimation is proceeded in one step with all the available measurements.

# LPV control

Several methods have been investigated for the design of LPV controllers adapted to robotic manipulators. The methods allow to guaranty some performance objective (H-infinity) over a variation domain.

## Based on dilated conditions

We have extended the dilated conditions introduced by Abkarian et al. in 2001 for LTI systems in order to synthesis LPV controllers and evaluated the result for the control of a 2-DOF manipulator. In order to remove the rational dependence of the scheduling parameters, the slack matrix used in the test is chosen as proportional to the inertia matrix. [4-HLB11]

## Based on descriptor model

The regular state-space models of the manipulators generally have a rational dependance with respect to the scheduling parameters. However, the descriptor formulation (M(q) dx/dt = …) can be obtained with a linear nature of the parameter dependance. Then, the tests for analysis and synthesis of descriptor systems has been adapted for the manipulator. [4-HLB10]

## Based on sum of squares (SOS) relaxations

Although the physical model has a rational dependance with respect to the scheduling parameters, it is possible to identified a model with polynomial dependance that has a very similar behavior. Then the SOS relaxations can be used in order to design analysis and synthesis tests based on the bounded real lemma. [1-HCMLxx], [5-HLB12]

# Funding

- PhD grant from French ministry of research (H. Halalchi, 2008-2011)
- PEPS CNRS IDRAC (2010-2011)

# Publications

- H. Halalchi, E. Laroche, G. Bara. Flexible-Link Robot Control Using a Linear Parameter Varying Systems Methodology,
*International Journal of Advanced Robotic Systems*, InTech, Vol. 11(46):1--12, March 2014. - D. Vizer, G. Mercere, E. Laroche, O. Prot. Linear fractional LPV model identification from local experiments using an H∞-based glocal approach, in
*Control-oriented modelling and identification: theory and practice*, Chap. 9, pp. 189-214, M. Lovera (Eds.), The Institution of Engineering and Technology, February 2014. - D. Vizer, G. Mercere, E. Laroche, O. Prot. LPV modeling and identification of a 2-DOF flexible robotic arm from local experiments using an H∞-based global approach, in
*Control-oriented modelling and identification: theory and practice*, Chap. 16, pp. 365-384, M. Lovera (Eds.), The Institution of Engineering and Technology, February 2014. - H. Halalchi, L. Cuvillon, G. Mercere, E. Laroche. Commande robuste des robots manipulateurs à flexibilités structurelles, in
*Structures flexibles - Applications à la manipulation robotique multi-échelle*, série Systèmes automatisés, Vol. Traité IC2, Chap. 11, pp. 359--389, M. Grossard, S. Régnier et N. Chaillet (Eds.), Hermès, 2013. - H. Halalchi, L. Cuvillon, G. Mercere, E. Laroche. Robust control of robotic manipulators with structural flexibilities, in
*Flexible Robotics*, Chap. 10, pp. 349--382, Mathieu Grossard, Stéphane Régnier and Nicolas Chaillet (Eds.), Wiley, July 2013. - D. Vizer, G. Mercere, O. Prot, E. Laroche. Combining analytic and experimental information for linear parameter-varying model identification: application to a flexible robotic manipulator,
*Periodica polytechnica*. Electrical engineering and computer science, Budapest University of Tecnology and Economics, Vol. 58(4):133-148, January 2014. - D. Vizer, G. Mercere, E. Laroche, O. Prot, M. Lovera. LPV/LFR model identification from local experiments: an H-infinity-based optimization technique,
*Conference on Decision and Control (CDC)*, Florence, Italy, December 2013. - G. Mercere, E. Laroche, O. Prot. Analytical Modelling and Grey-box Identification of a Flexible Arm using a Linear Parameter-varying Model,
*16th IFAC Symposium on System Identification*, Brussels, Belgium, July 2012. - H. Halalchi, G. Bara, E. Laroche. Observer-Based Controller Synthesis for LPV Descriptor Systems Using Dilated LMI Conditions,
*IEEE Multi-Conference on Systems and Control (MSC 2011)*, Denver, CO, United States, September 2011. - H. Halalchi, E. Laroche, G. Bara. Output feedback LPV control strategies for flexible robot arms,
*18th IFAC World Congress*, Milan, Italy, August 2011. - H. Halalchi, E. Laroche, G. Bara. A Polynomial LPV Approach for Flexible Robot End-Effector Position Controller Analysis,
*50th IEEE Conference on Decision and Control and European Control Conference*, Orlando, FL, United States, December 2011. - G. Mercere, E. Laroche, M. Lovera. Identification of a flexible robot manipulator using a linear parameter-varying descriptor state-space structure,
*IEEE Conference on Decision and Control, Orlando (FL)*, United States, December 2011. - H. Halalchi, G. Bara, E. Laroche. LPV Controller Design for Robot Manipulators Based on Augmented LMI Conditions with Structural Constraints,
*4th IFAC Symposium on System, Structure and Control*, Ancona, Italy, September 2010. - H. Halalchi, E. Laroche, G. Bara. LPV modeling and control of a 2-DOF robotic manipulator based on descriptor representation,
*11th Pan-American Congress of Applied Mechanics (PACAM XI)*, Foz do Iguaçu, PR, Brazil, January 2010. - H. Halalchi, E. Laroche, G. Bara. Analyse de performance d'asservissement opérationnel pour manipulateurs flexibles suivant une approche LPV polynomiale,
*Conférence Internationale Francophone d'Automatique (CIFA 2012)*, Grenoble, France, July 2012. - H. Halalchi, E. Laroche, G. Bara. Contrôle LPV par retour de sortie pour les manipulateurs flexibles,
*Journées doctorales du GdR MACS*, Marseille, France, June 2011. - E. Laroche, G. Mercere, H. Halalchi. Modélisation et identification LPV d'un manipulateur flexible,
*Journées Identification et Modélisation Expérimentale*, Douai, France, April 2011.