Automatica 38 (2002) 1783–1790 www.elsevier.com/locate/automatica Brief Paper Second-order sliding-mode control of container cranes  Giorgio Bartolini * , Alessandro Pisano, Elio Usai Dipartimento di Ingegneria Elettrica ed Elettronica, Universit a di Cagliari piazza d’Armi, I-09123 Cagliari, Italy Received 24 November 2000; received in revised form 24 July 2001; accepted 2 May 2002 Abstract Moving a suspended load along a pre-specied path is not an easy task when strict specications on the swing angle and on the transfer time need to be satised. Intuitively, minimizing the cycle time and the load swing are conicting requirements, and their satisfaction requires proper control actions, especially if some uncertainties in the system dynamics are present. In this paper we propose a simple control scheme, based on second-order sliding modes, which guarantees a fast and precise load transfer and the swing suppression during the load movement, despite of model uncertainties and unmodeled dynamic actuators. Such controller has been tested on a laboratory-size model of the crane, and some experimental results are reported. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Container cranes; Second-order sliding modes; Uncertain systems; Nonlinear systems 1. Introduction Inmanyindustrialaswellascivilengineeringenvironments, safety and economical constraints demand a fast and pre- cise transfer of suspended heavy loads over long distances, minimizing the load oscillation and the operation time. Human drivers, often aided by automatic anti-sway sys- tems, are always involved, and the resulting performance, in terms of swiftness and safety, heavily depends on their experience and capability. For this reason, a growing inter- est is arising about the design of automatic control systems for container cranes. The relevant control strategy is often designed in two-stages: o-line path planning, in accordance with some optimization criteria, and on-line path tracking. In partic- ular, optimal control techniques have been widely used to address the path planning problem (Auernig & Troger, 1987; Beeston, 1969; Hamalainen, Marttinen, Baharova, & Virkkunen, 1995; Sakawa & Shindo, 1982; Hong, Park, & Lee, 2000). The reference trajectories were chosen to min- imize some specic indices, linked to the swing angle and its derivative (Sakawa & Shindo, 1982), or to the energy  This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Tsutomu Mito under the direction of Editor Mituhiko Araki. ∗ Corresponding author. Tel.: +39-0706-755869; fax: +39-0706- 755900. E-mail address: giob@dist.unige.it (G. Bartolini). consumption, which is claimed to be meaningful with re- spect to system stresses such as oscillations and non-smooth motions (Hamalainen et al., 1995). In Hong et al. (2000) an interesting combination of time-optimal control (for the path planning), and variable structure control (for the path tracking), can be found, together with experimental results. Other approaches are based on the knowledge of a linear, parameter varying, crane model, obtained by means of a suitable time scaling (Virkkunen, 1990; Giua, Seatzu, & Usai, 1999) that allows for the use of adaptive pole-placement control techniques (Virkkunen, 1990), gain scheduling (Corriga, Giua, & Usai, 1998) or Lyapunov-equivalence-based observer= controller design (Giua et al., 1999). While the usual goal is to achieve zero-swing only at the end of the transport, in Moustafa and Ebeid (1988) an eective approach to obtain the swing suppression also during the transfer of the load was presented. This is a desir- able property since the eect of environmental disturbances, such as wind gusts, must be necessarily taken into account. The controller in Moustafa and Ebeid (1988) was based on local linearization, and, therefore, it requires the perfect knowledge of the system model. On the contrary, in this paper we address and solve the same problem assuming that the actual system parameters (including those of the actu- ators) are uncertain but belonging to a known compact do- main; thus, the use of robust control techniques appears to be motivated. 0005-1098/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII:S0005-1098(02)00081-X