Control of a flexible joint robot manipulator via a non-linear control-observer scheme J. DE LEON-MORALEST*, J. G. ALVAREZ-LEAL$, R. CASTRO-LINARESS and JA. ALVAREZ-GALLEGO3 A non-linear controller-observer scheme for the output tracking of a class of non-linear singularly perturbed systems based on a two-time scale sliding-mode technique and a high gain estimator, is presented. An analysis or stability ot' thc resultant closed-loop system is given. The proposed scheme is applied to the model of a two degrees of fr-ezdom llesible joint robot to show the controller-observer methodology proposed. 1. Introduction During the last few years, considerable research efyorts have been directed toward the control problem of flcxiblc joint robots. Various techniques such as feed- back linearization, singular perturbations, sliding-mode, pass~vity and adaptive techniques have been proposed to tackle the problem (see, for example, Marino and Spong 1986. Spong 1990, Kokotovic et al. 1993, Battilotti and Lanaii 1995, Brogliato et al. 1995, 1998, Hernandez and Ba-bot 1996). The aim of this paper is to design an observer-based controller for tracking the desired reference signal for Aex~blemanipulators combming the advantages of the singular perturbation methods and sliding modes tech- mques. It IS well known that many physical systems involve dynam~c phenomena occurring in different time scales ,ind the ilcxible joint manipulator is a representative ex- ample of them. Typically, these systems can be modelled usmg a singular perturbation approach which allows us to obtain subsystems of reduced dimension in order to design control laws and analyse their stability (see, for example, Kokotovic st al. 1993). The sliding-mode control of rigid robotic manipula- tors as a robust approach has attracted a number of researchers (see Nathan and Singh 1987, De Carlo rt ul. 1988, Utkin 1992, Stepanenko and Su 1993). More- over. a sliding controller is characterized as a high-speed switching controller that provides a robnst means of controlling non- linear systems by forcing the trajectories Recelved 18 November 1998. Revised 8 October 19% and 12 Tune 2000. W~ithor for correspondence. e-mall: jleon@,ccr.dsi.uanl.mx t Univers~dad Aut@noma de Nuevo Le6n, Programa Doctoral en Ingen~eria, Apdo. Postal 148-F, 66451 San N~colas de Los Gawa. N. L., Mexico. Inst~tuto Tecnolbgico de Saltillo, Apdo. Postal 58, Suc. C. 3380 Saltillo, Coahu~la, Mexico b CINVESTAV-IPN, Depto. Ingenieria EICctrica, Apdo. Postal 14-740, 07300 Mex~co, DF. to reach a sliding manifold in finite time and stay on tlic manifold for all time. Due to switching bcliavio~ir of the controller some theoretical and practical problems rise. On the one hand, the controller contains ;I discon- tinuous non-linear term for which tlic cxistc~~cc and uniqueness of solutions should be examined. On tl~c other hand, the observer design for flexible joint nianip- ulators is an important problem in robot control theory and of great practical importance as well. 111 several works, the control law requires measurclnent ol' tlic joint position and the joint velocity. Hnwcvcr, the joint velocity is often contaminated by noisc which reduces the performance of the controller. In order to avoid these problems, some interesting rcsults on the observer problem Sor these robots are rcpoi-tcd, ['or cs- ample, in Tomei (1990) and Nicosia ct rll. (19SS). More recently, in Hernandez and Barbot (1996). a sliding observer-based reedback control law, using sing~~lur p c ~ turbed methods and sliding-modes, has bccn designcd for a class of non- linear systems which admits a singular perturbation representation for tracking desired tinic- varying trajectories for flexible manipulators. 7'he flexible joint manipulator model can be split into two subsystems: the slow and fast subsystems, wlicrc the slow variables are the link positions and vclocitics and the fast variables are the elastic forces aod their time derivatives. In Hernandez and Barbot (1 W6) lhc Join[ positions and the elastic force, which rcl~rcscnr the out - put of the slow subsystem and the output of the Sast subsystem, respectively, are measured. For that, tl~c authors propose a sliding observer to cstimatc the unmeasurable variables. Assuming that the output variable is the only one that is available for measurement in a non-linear ~ystciii~ in this paper a controller-observer scheme is prcyoscd to make that variable track a given reference signal. 'I'he controller is based on a sliding mode tcchnicl~~c rcccntly developed in Alvarez and Silva (1997) to stabilize a class of non-linear singular perturbed systems. 'I'he main dflercnce of the present work with rcspcct lo that oS