The motion planning problem and exponential stabilisation of a heavy chain. Part I

Abstract

Abstract A model of a heavy chain system with a punctual load (tip mass) in the form of a system of partial differential equations is interpreted as an abstract semigroup system on a Hilbert state space. Our aim is to solve the output motion planning problem of the same nature as in the case of an unloaded heavy chain (Grabowski, P. (Citation2003), ‘Abstract Semigroup Model of Heavy Chain System with Application to a Motion Planning Problem’, in Proceedings of 9th IEEE International Conference: Methods and Models in Automation and Robotics, 25–28 August, Międzyzdroje, Poland, pp. 77–86 (IS1-2-3.PDF)). In order to solve this problem we first analyse its well-posedness and some basic properties. Next, we solve the output motion planning problem using a substitute of the inverse of the input–output operator represented in terms of the Laplace transforms. A problem of exponential stabilisation is also formulated and solved using a stabiliser of the colocated type. The exponential stabilisation is proved using the method of Lyapunov functionals combined with some frequency-domain tools. The method of Lyapunov functionals can be replaced by the spectral or exact controllability approach as shown in the second part (Grabowski, P. (Citation2008), ‘The Motion Planning Problem and Exponential Stabilisation of a Heavy Chain. Part II’, Opuscula Mathematica, 28 (Citation2008) (Special issue dedicated to the memory of Professor Andrzej Lasota), 481–505) of the present article. A laboratory setup which allows verification of the results in practice is described in detail. Its dynamical model is used as an example to illustrate the theoretical results. †Dedicated to Frank M. Callier on the occasion of his 65th birthday. Keywords: infinite-dimensional control systemssemigroupsmotion planning problemexponential stabilisationLyapunov functionals Notes †Dedicated to Frank M. Callier on the occasion of his 65th birthday. A1. Where, in addition, the identity Abramowitz and Stegun (Citation1984, 9.1.3) has been used to eliminate a Hankel function normally appearing in Abramowitz and Stegun (Citation1984, 9.6.4). A2. This fact can be also easily concluded from the result of Triggiani (Citation1989). A3. We remark that the authors of d'Andréa-Novel et al. (Citation1994) use the same orientation of the spatial variable as ours. A4. To be more precise, when the Nyquist plot encircles the disk D. Notes 1. NSK Ltd. is the Japanese company producing hardware for the control of mechanical systems. 2. Currently a digital video camera recording the chain positions was installed. 3. They may be obtained by comparing (Equation6.3) with Curtain et al. (Citation2003, (5.4)). 4. The formula (Thull et al. Citation2005, (36), p. 405) cannot define a scalar product on a state space proposed by the authors. This is because it employs the functional of taking derivative at a point of a H1(0, L)-function, but such a linear functional is neither everywhere defined nor continuous on H1(0, L). 5. Nowadays, the access to a video movie via http://www.LSR.uni-saarland.de/rd/crane06.htm, announced in Thull et al. (Citation2006, p. 661), has been transferred http://cds.acin.tuwien.ac.at/fileadmin/cds/data/video/Kette.mpeg