Automatica 45 (2009) 1020–1025 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Brief paper Adaptive motion coordination: Using relative velocity feedback to track a reference velocity ✩ He Bai a,∗ , Murat Arcak a,b , John T. Wen a a Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA b Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720, USA article info Article history: Received 29 November 2007 Received in revised form 18 August 2008 Accepted 7 November 2008 Available online 20 January 2009 Keywords: Cooperative control Adaptive design Passivity Parameter convergence abstract We study a coordination problem where the objective is to steer a group of agents to a formation that translates with a prescribed reference velocity. In Bai et al. [Bai, H., Arcak, M., & Wen, J. (2008). Adaptive design for reference velocity recovery in motion coordination. Systems and Control Letters, 57(8), 602–610.] we considered the situation where the reference velocity information is available only to a leader, and developed a decentralized adaptive design that uses relative position feedback. Although Bai et al. (please see above reference) guaranteed the desired formation, it did not ensure tracking of the reference velocity with the exception of special cases. We now propose a new adaptive redesign that guarantees tracking of the reference velocity by incorporating relative velocity feedback in addition to relative position feedback. Published by Elsevier Ltd 1. Introduction Growing research in motion coordination has led to significant results in formation control, consensus, deep-space flying, swarm- ing, etc. The main challenge in cooperative control is designing de- centralized control laws that depend on relative information, to achieve desired group behaviors. The design techniques developed so far employ Lyapunov analysis and potential function methods (Gazi, 2005b; Ögren, Fiorelli, & Leonard, 2004; Tanner, Jadbabaie & Pappas, 2007), matrix analysis (Jadbabaie, Lin & Morse, 2003), graph theoretic results (Fax & Murray, 2004; Ren, Moore & Chen, 2007) and nonlinear servomechanism (Gazi, 2005a). The recent study in Arcak (2007) introduced passivity as a unifying design framework for cooperative control problems, such as consensus and formation control. This passivity framework ✩ This work is partially supported by the Air Force Office of Scientific Research under award No. FA9550-07-1-0308. The work of the third author is partially supported by the Center for Automation Technologies and Systems (CATS) under a block grant from the New York State Office of Science, Technology, and Academic Research (NYSTAR) and by the Outstanding Overseas Chinese Scholars Fund of Chinese Academy of Sciences (No. 2005-1-11). This paper was partially presented at 2008 American Control Conference, Seattle, Washington, USA. This paper was recommended for publication in revised form by Associate Editor Alessandro Astolfi under the direction of Editor Hassan K. Khalil. ∗ Corresponding address: CII 8123, 110 8th Street, Troy, NY 12180, USA. Tel.: +1 518 276 2973; fax: +1 518 276 4897. E-mail addresses: baih@rpi.edu (H. Bai), arcak@eecs.berkeley.edu (M. Arcak), wenj@rpi.edu (J.T. Wen). encompasses several potential function based designs in the literature, such as Ögren et al. (2004), and offers further design flexibility to address complex agent dynamics such as rigid body models studied by Bai, Arcak and Wen (2007). Arcak (2007), as well as earlier results such as Ögren et al. (2004), assumed that the reference velocity is available to each agent. This assumption is relaxed in Bai, Arcak, and Wen (2008) to the situation where only the leader possesses the reference velocity while the other agents estimate this information by employing a decentralized adaptive design. Although this basic adaptive design guaranteed the convergence to the desired formation, it did not ensure tracking of the reference velocity except in two special cases: First, when the reference velocity is constant and, second, in the special case of an agreement design in which the reference velocity satisfies a persistency of excitation property. In this paper, we propose an augmented adaptive redesign to guarantee tracking of the reference velocity without the restrictions of Bai et al. (2008) discussed above. The main idea in our augmented design is to ensure that the relative velocities between agents converge to zero, thereby guaranteeing that all agents converge to the reference velocity. By including the relative velocity feedback in our redesign, we recover the stability result of Bai et al. (2008) while achieving tracking of the reference velocity in addition. Another study that employs an estimation algorithm to compensate for limited communication between agents has been reported in Sepulchre, Paley, and Leonard (2008). The study of Sepulchre et al. (2008), however, addresses identical 0005-1098/$ – see front matter. Published by Elsevier Ltd doi:10.1016/j.automatica.2008.11.008