Adaptive input shaping for single-link flexible manipulators using an algebraic identification Emiliano Pereira a,n , Juan R. Trapero b , Iva ´n M. Dı ´az c , Vicente Feliu c a Escuela Polite´cnica Superior, Universidad de Alcala ´ de Henares, Ctra. Madrid-Barcelona, Km. 33,600, 28805 Alcala ´ de Henares, Madrid, Spain b Facultad de Ciencias Quı ´micas, Universidad de Castilla-La Mancha, Av. Camilo Jose´ Cela s/n, 13071 Ciudad Real, Spain c Escuela Te´cnica Superior de Ingenieros Industriales, Universidad de Castilla-La Mancha, Av. Camilo Jose Cela s/n, 13071 Ciudad Real, Spain article info Article history: Received 22 March 2010 Accepted 9 October 2011 Available online 26 October 2011 Keywords: Identification algorithms Adaptive control Open-loop control systems Flexible arms Trajectories and Monte Carlo simulation abstract This work proposes an adaptive control scheme applied to single link-flexible manipulators, which combines a feedback controller of the joint angle with an adaptive input shaper updated by an algebraic non-asymptotic identification. The feedback controller is designed to guarantee trajectory tracking of the joint angle, simplifying thus the input shaper, which can be designed for the arm dynamics only. The input shaper is updated by an algebraic identification of the natural frequency corresponding to the first vibration mode of the arm. In addition, the influence of the assumptions adopted to derive the algebraic identification on the performance of the estimation is studied. Finally, the proposed adaptive control strategy is implemented in practice. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction The use of flexible manipulators presents many advantages with respect to their rigid counterparts. For instance, the aero- space industry demands lighter structures that can be driven using small amount of energy. Weight reduction usually increases the flexibility of structures. Other examples can be found in structures working in fragile environments or with humans, mitigating thus the damage of a possible collision. That is, the flexibility absorbs most of the energy of possible impacts. How- ever, flexibility makes more difficult to obtain analogous perfor- mance requirements, such as high speed operation or precise positioning, as compared with rigid manipulators. In order to address control objectives, such as tip position accuracy and suppression of residual vibration, many control techniques have been applied to flexible robots (see, for instance, the surveys Benosman & Vey, 2004; Dwivedy & Eberhard, 2006; Feliu, 2006 or the book of Tokhi & Azad, 2008). One of these techniques consists of generating a reference command that does not excite the vibration modes of the system, and therefore, the final position is achieved without residual vibration. The most used technique employed to generate com- mand reference is based on filtering the desired trajectory in real time by using an input shaper (IS). An IS is a particular case of a finite impulse response filter that obtains the command reference by convolving the desired trajectory with a sequence of impulses (filter coefficients) (Singer & Seering, 1990; Smith, 1958). Since the work of Singer and Seering (1990), many applications have appeared, such as: spacecraft field (Tuttle & Seering, 1997), cranes and structures like cranes (see applications and performance comparisons in Huey, Sorensen, & Singhose, 2008), manipulation of liquid containers (Feddema et al., 1997) or nanopositioners (Jordan, 2002). One of the main problems of implementing an IS in the industry is to deal with system uncertainties produced by model errors or changes in the system parameters. The proposed solu- tions can be classified into robust, learning or adaptive input shaping approaches. An adaptive input shaper (AIS) updates the parameters of an IS (or even a robust IS) without considering previous manoeuvres (see, for example, Pereira, Trapero, Dı ´az, & Feliu, 2009), in such a way that this option is more suitable than learning approaches when non-repetitive manoeuvres are involved (Park, Chan, Park, & Lee, 2006; Park & Chang, 2001). The performance of an AIS depends on the identification procedure chosen. An AIS may be developed in the frequency domain (Tzes & Yurkovich, 1993), or in the time domain (Bodson, 1998; Cutforth & Pao, 2004; Pereira et al., 2009; Rhim & Book, 2001, 2004). In order to derive the time intervals of the input shaper impulses in an on-line fashion, Tzes and Yurkovich (1993) employed the time-varying transfer function estimation in the frequency domain. Nonetheless, this technique implies a high numeric burden and requires a considerable number of periods to obtain the system parameters with sufficient precision. This fact has motivated the use of identification techniques in time Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/conengprac Control Engineering Practice 0967-0661/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2011.10.004 n Corresponding author. Tel.: þ34 91 885 67 11; fax: þ34 926 29 53 61. E-mail address: Emiliano.Pereira@uah.es (E. Pereira). Control Engineering Practice 20 (2012) 138–147