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Bifurcations and symmetry in two optimal formation control problems for mobile robotic systems
- Baoyang Deng, Michael O'Connor, Bill Goodwine
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This paper studies bifurcations in the solution structure of an optimal control problem for mobile robotic formation control. In particular, this paper studies a group of mobile robots operating in a two-dimensional environment. Each robot has a predefined initial state and final state and we compute an optimal path between the two states for every robot. The path is optimized with respect to two factors, the control effort and the deviation from a desired “formation,” and a bifurcation parameter gives the relative weight given to each factor. Using an asymptotic analysis, we show that for small values of the bifurcation parameter (corresponding to heavily weighting the control effort) a single unique solution is expected, and that as the bifurcation parameter becomes large (corresponding to heavily weighting maintaining the formation) a large number of solutions is expected. Between the asymptotic extremes, a numerical investigation indicates a solution bifurcation structure with a cascade of increasing numbers of solutions, reminiscent, but not the same as, period-doubling bifurcations leading to chaos in dynamical systems. Furthermore, we show that if the system is symmetric, the bifurcation structure possesses symmetries, and also present a symmetry-breaking example of a non-holonomic system. Knowledge and understanding of the existence and structure of bifurcations in the solutions of this type of formation control problem are important for robotics engineers because common optimization approaches based on gradient-descent are only likely to converge to the single nearest solution, and a more global study provides a deeper and more comprehensive understanding of the nature of this important problem in robotics.
nine - Citizenship, community and participation in small towns: a case study of regeneration partnerships
- Edited by Rob Imrie, Mike Raco
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- Book:
- Urban Renaissance?
- Published by:
- Bristol University Press
- Published online:
- 20 January 2022
- Print publication:
- 21 May 2003, pp 181-204
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Summary
Introduction
Small towns, regeneration policy and the assumption of community
Small or market towns with populations of between 2,000 and 20,000 people are a neglected part of the urban studies literature, a point confirmed by a recent report by the Countryside Agency, which concluded that “data and knowledge about market towns is generally weak” (2002a, p 1). Indeed, in many ways, research on small towns has been hampered by an academic division of labour, which categorises objects of enquiry as either ‘urban’ or ‘rural’. The disciplines of Geography and Sociology have both recognised sub-disciplines in Urban Geography, Rural Geography, Urban Sociology and Rural Sociology, complemented by interdisciplinary work in urban or rural studies. Specialised journals and research groups follow this pattern. Small towns fall uneasily between the two areas – not large enough to be readily admitted as ‘urban’, but too large to fall within the classic agricultural and village foci of rural studies. Urban policy work in particular has concentrated on larger urban areas and conurbations, and almost all empirical examples of urban policies in practice are drawn from cities such as London, Manchester, Liverpool, Leeds, Glasgow and Birmingham. This is not just a question of scale: empirical studies of urban policy in these conurbations often examine neighbourhood or estatebased schemes that target populations of less than 5,000. More precisely, then, it is a question of small towns as a category falling outside the lens of most urban-focused research.
This is particularly unfortunate at the present juncture, when small towns are viewed as critical sites for regeneration strategies that serve their own communities and those of surrounding areas, and when they are also seen as places where community engagement and involvement in such strategies can be easily fostered. A new policy focus on small towns was heralded by Labour's Rural White Paper for England, Our countryside: The future (DETR and MAFF, 2000). For the first time in a rural White Paper, a chapter was dedicated to ‘market towns’. In setting out a coherent strategy for small-town regeneration, the White Paper drew together a number of initiatives targeted at small towns that were already operated by the Countryside Agency and Regional Development Agencies (RDAs), as well as by the Welsh Development Agency.